{"id":162,"date":"2021-03-01T08:52:19","date_gmt":"2021-03-01T13:52:19","guid":{"rendered":"http:\/\/pressbooks.library.upei.ca\/statics\/?post_type=chapter&#038;p=162"},"modified":"2025-07-23T14:49:27","modified_gmt":"2025-07-23T18:49:27","slug":"right-hand-rule","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.upei.ca\/statics\/chapter\/right-hand-rule\/","title":{"raw":"3.1 Right Hand Rule","rendered":"3.1 Right Hand Rule"},"content":{"raw":"Before we can analyze rigid bodies, we need to learn a little trick to help us with the cross product called the 'right-hand rule'. We use the right-hand rule when we have two of the axes and need to find the direction of the third.\r\n<div><\/div>\r\n<div>This is called a right-orthogonal system. The '<em>orthogonal'<\/em> part means that the three axes are all perpendicular to each other, and the' <em>right'<\/em> part means that [latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=\\underline{\\hat{k}}[\/latex], hence the right hand rule. Remember these from section 1.5?<\/div>\r\n<div>\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\r\n \t<li>[latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=\\underline{\\hat{k}}[\/latex]<\/li>\r\n \t<li>[latex]\\underline{\\hat{j}}\\times\\underline{\\hat{k}}=\\underline{\\hat{i}}[\/latex]<\/li>\r\n \t<li>[latex]\\underline{\\hat{k}}\\times\\underline{\\hat{i}}=\\underline{\\hat{j}}[\/latex]<\/li>\r\n \t<li>[latex]\\underline{\\hat{j}}\\times\\underline{\\hat{i}}=-\\underline{\\hat{k}}[\/latex]<\/li>\r\n \t<li>[latex]\\underline{\\hat{k}}\\times\\underline{\\hat{j}}=-\\underline{\\hat{i}}[\/latex]<\/li>\r\n \t<li>[latex]\\underline{\\hat{i}}\\times\\underline{\\hat{k}}=-\\underline{\\hat{j}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe opposite of the right-orthogonal system is the left-orthogonal system where [latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=-\\underline{\\hat{k}}[\/latex]. We don't use that one!\r\n\r\nThere are two ways to do the right hand rule, and they take practice to conceptually understand, but this will make solving problems much quicker. You're going to use your fingers and thumb to represent the x, y, and z axes.\r\n<h1><strong>3.1.1 The Whole-Hand Method<\/strong><\/h1>\r\n<div>In the following description, A x B = C, so for the coordinate frame, X x Y = Z ([latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=\\underline{\\hat{k}}[\/latex]). Your fingers go in the direction of X, then you\u00a0 bend them 90 degrees to point towards Y, and your thumb is in the direction of Z.<\/div>\r\n<div><\/div>\r\n<div class=\"textbox\">\r\n\r\nThe direction of the cross product vector A x B is given by the right-hand rule for the cross product of two vectors. To apply this right-hand rule, extend the fingers of your right hand so that they are pointing directly away from your right elbow. Extend your thumb so that it is at\r\nright angles to your fingers.\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0519.jpg\" alt=\"A drawing of a right hand with the thumb extended upward and fingers pointing forward.\" class=\"aligncenter wp-image-257\" width=\"253\" height=\"208\" \/>\r\n\r\nKeeping your fingers aligned with your forearm, point your fingers in the direction of the first vector (the one that appears before the \u201c\u00d7\u201d in the mathematical expression for the cross product; e.g. the A in A x B ).\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520.jpg\" alt=\"A right-hand diagram showing the thumb pointing up (B) and fingers pointing right (A).\" class=\"aligncenter wp-image-258\" width=\"293\" height=\"254\" \/>\r\n\r\nNow rotate your hand, as necessary, about an imaginary axis extending along your forearm and along your middle finger, until your hand is oriented such that, if you were to close your fingers, they would point in the\u00a0direction of the second vector.\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521.jpg\" alt=\"A right-hand diagram with the thumb pointing out of the page, fingers pointing right (A), and palm facing upward with vector B.\" class=\"aligncenter wp-image-259\" width=\"326\" height=\"190\" \/>\r\n\r\nYour thumb is now pointing in the direction of the cross product vector. C = A x B. The cross product vector C is always perpendicular to both of the vectors that are in the cross product (the A and the B in the case at hand). Hence, if you draw them so that both of the vectors that are in the cross product are in the plane of the page, the cross product vector will always be perpendicular to the page, either straight into the page, or straight out of the page. In the case at hand, it is straight out of the page.\r\n\r\nWhen we use the cross product to calculate the torque due to a force F whose point of application has a position vector r, relative to the point about which we are calculating the torque, we get an axial torque vector \u03c4. To determine the sense of rotation that such a torque vector would correspond to, about the axis defined by the torque vector itself, we use the Right Hand Rule for Something Curly Something Straight. Note that we are calculating the torque with respect to a point rather than an axis\u2014the axis about which the torque acts comes out in the answer.\r\n\r\n&nbsp;\r\n\r\nSource: <span>Jeffrey W. Schnick <\/span><a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7<\/a> pages 135\u2013137\r\n\r\n<\/div>\r\nThe hardest part of the right-hand rule is imagining the different axes and envisioning how they are perpendicular to each other. <img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-1024x218.png\" alt=\"Diagrams showing the x, y, and z axes with right angle symbols to illustrate perpendicularity between each axis pair.\" class=\"aligncenter wp-image-1126 size-large\" width=\"1024\" height=\"218\" \/>\r\n\r\nTry this one in 2d and 3d. Imagine (or draw) the right-angle symbols (Answer will be in a few steps)<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM.png\" alt=\"Two diagrams showing right-handed coordinate systems with x, y, and z axes.\" class=\"aligncenter wp-image-1127\" width=\"306\" height=\"110\" \/>\r\n<h2><strong>Example 1:<\/strong><\/h2>\r\nUsing these x and y, let's use the right-hand rule to find the direction of z.\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-1024x304.png\" alt=\"Two 3D axis diagrams showing right-angle symbols between x, y, and z axes in different colors. \" class=\"aligncenter wp-image-1130\" width=\"579\" height=\"172\" \/>\r\n\r\nHere are the steps you can follow:\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-1024x748.png\" alt=\"A 4-step visual guide demonstrating the right-hand rule for determining 3D coordinate directions using photos of a hand and labeled axes.\" class=\"aligncenter wp-image-1131 size-large\" width=\"1024\" height=\"748\" \/>\r\n\r\n&nbsp;\r\n<h3>Example 2:<\/h3>\r\nSometimes you will need to flip your hand 180 degrees to find which way lets you point your fingers in the y direction, for example:\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-1024x1005.png\" alt=\"A step-by-step guide using hand photos and 3D coordinate diagrams to explain the right-hand rule and identify the correct direction of the z-axis.\" class=\"aligncenter wp-image-1150 size-large\" width=\"1024\" height=\"1005\" \/>\r\n\r\n&nbsp;\r\n<h3>Example 3:<\/h3>\r\nIt's important for you to be able to envision how the axes are perpendicular. Now practice using the right hand rule if you are trying to find x.\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-1024x263.png\" alt=\"A visual explanation of using the right-hand rule to find the x-axis direction when z and y are known, showing hand positioning and a 3D coordinate system.\" class=\"aligncenter wp-image-1151 size-large\" width=\"1024\" height=\"263\" \/>\r\n<h3>Your Turn!<\/h3>\r\nKeep going with these examples. The rules stay the same: thumb towards z, curled fingers towards y, extended fingers towards x. Find the missing axis:<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-1024x207.png\" alt=\"Five differently colored 3D coordinate axes showing various orientations of the x, y, and z directions.\" class=\"aligncenter wp-image-1152 size-large\" width=\"1024\" height=\"207\" \/>\r\n<p style=\"text-align: center\">.<\/p>\r\n<p style=\"text-align: center\">.<\/p>\r\n<p style=\"text-align: center\">Did you do it?<\/p>\r\n<p style=\"text-align: center\">.<\/p>\r\n<p style=\"text-align: center\">.<\/p>\r\n<p style=\"text-align: center\">.<\/p>\r\n<p style=\"text-align: center\">Here are the answers:<\/p>\r\n<p style=\"text-align: center\">.<\/p>\r\n<p style=\"text-align: center\">.<\/p>\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-1024x248.png\" alt=\"Five colored 3D coordinate systems with green arrows indicating the correct direction of the third axis using the right-hand rule.\" class=\"aligncenter wp-image-1153 size-large\" width=\"1024\" height=\"248\" \/>\r\n<h1>3.1.2 Right Hand Rule and Torque<\/h1>\r\nThe third way to calculate torque, as was alluded to in Section 1.6, is to use the right-hand rule to identify the axis of rotation. The first way (the scalar method) uses | <span><span style=\"text-decoration: underline\"><strong>M<\/strong><\/span> |<\/span> = |<span style=\"text-decoration: underline\"><strong>r<\/strong><\/span>| |<span style=\"text-decoration: underline\"><strong>F<\/strong><\/span>| sin \u0398, and often the angle between the position vector and force is 90 degrees. The vector method is for more complicated situations and uses the cross product <span style=\"text-decoration: underline\"> <strong>r<\/strong><\/span> x <span style=\"text-decoration: underline\"><strong>F<\/strong><\/span><strong> = <\/strong><span style=\"text-decoration: underline\"><strong>M<\/strong><\/span><strong>. <\/strong>The third method finds the scalar value separately, then uses the right-hand rule to find the direction (positive or negative along the third axis).\r\n<ul>\r\n \t<li>Point your fingers in the direction of <em><span style=\"text-decoration: underline\">the perpendicular part<\/span><\/em> of the position vector <span style=\"text-decoration: underline\"><strong>r<span><\/span><\/strong><\/span> (as you would for x)<\/li>\r\n \t<li>Curl them towards the direction of the Force vector <span style=\"text-decoration: underline\"><strong>F<\/strong><\/span> (as you would for y)<\/li>\r\n \t<li>Your thumb is in the direction of the moment <span style=\"text-decoration: underline\"><strong>M<\/strong><\/span> that results from the force (as for z)<\/li>\r\n<\/ul>\r\nThe following will help you understand what is meant by: <em>the perpendicular part <\/em>of the position vector:\r\n<div class=\"textbox\">\r\n<div class=\"page\" title=\"Page 132\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<span>The torque <\/span><span>\u03c4 <\/span><span>can be expressed as the cross product of the position vector <\/span><span>r <\/span><span>for the point of application of the force, and the force vector <\/span><span style=\"text-decoration: underline\"><strong>F<\/strong><\/span> <span>itself: <span style=\"text-decoration: underline\"><strong>r<\/strong><\/span> x <span style=\"text-decoration: underline\"><strong>F<\/strong><\/span><strong> =\u00a0<\/strong><span style=\"text-decoration: underline\"><strong>M<\/strong><\/span><\/span>\r\n\r\n<span>Before we begin our mathematical discussion of what we mean by the cross product, a few words about the vector <strong>r <\/strong>are in order. It is important for you to be able to distinguish between the position vector <strong>r <\/strong>for the force, and the moment arm, so we present them below in one and the same diagram. We use the same example that we have used before: <span style=\"text-decoration: underline\"><img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-1024x579.png\" alt=\"An irregular shaped object with a labeled axis of rotation at point O and a force F applied at a distant point.\" class=\"aligncenter wp-image-1155\" width=\"436\" height=\"247\" \/><\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<span>in which we are looking directly along the axis of rotation (so it looks like a dot) and the force lies in a plane perpendicular to that axis of rotation. We use the diagramatic convention that, the point at which the force is applied to the rigid body is the point at which one end of the arrow in <\/span><span>the diagram touches the rigid body. Now we add the line of action of the force and the moment arm r\u22a5 to the diagram, as well as the position vector r of the point of application of the force. <img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-1024x685.png\" alt=\"A diagram showing an object with force vec F, position vector vec r, moment arm r_perpendicular, and the line of action of the force labeled.\" class=\"aligncenter wp-image-1156\" width=\"441\" height=\"295\" \/><\/span><span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"page\" title=\"Page 133\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<span>The moment arm can actually be defined in terms of the position vector for the point of application of the force. Consider a tilted <\/span><span>x<\/span><span>-y coordinate system, having an origin on the axis of rotation, with one axis parallel to the line of action of the force and one axis perpendicular to the line of action of the force. We label the <\/span><span>x <\/span><span>axis <\/span><span>\u2534 <\/span><span>for \u201cperpendicular\u201d and the y axis <\/span><span>|| <\/span><span>for \"parallel\".<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM.png\" alt=\"Diagram showing a force vector vec F, position vector vec r, and red dashed lines representing parallel and perpendicular directions from the axis of rotation.\" class=\"aligncenter wp-image-1157\" width=\"354\" height=\"360\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"page\" title=\"Page 134\">\r\n<div class=\"section\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<span>Now we break up the position vector <\/span><span>r <\/span><span>into its component vectors along the \u2534 (perpendicular) and || (parallel) axes. <\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"section\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM.png\" alt=\"Diagram showing force vec F, position vector vec r, and its components r_perpendicular and r_parallel with dashed red lines indicating perpendicular and parallel directions.\" class=\"aligncenter wp-image-1158\" width=\"343\" height=\"284\" \/>\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<span>From the diagram it is clear that the moment arm <\/span><span>r <\/span><span>is just the magnitude of the component <\/span><span>\u2534 <\/span><span>vector, in the perpendicular-to-the-force direction, of the position vector of the point of application of the force. <\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"layoutArea\">\r\n<div class=\"column\"><\/div>\r\n<\/div>\r\n<div>Source: <span>Calculus\u00a0Based Physics, Jeffrey W. Schnick,\u00a0<\/span><a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7\" class=\"\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7<\/a> pages 132\u2013137<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\nYou use the right-hand rule twice during this method to find the vector. First, to determine the coordinate frame and again to see in which direction the torque is aligned. Then you multiply by the magnitude of the perpendicular portion of the position vector (r\u22a5 or the \"moment arm\") and the magnitude of the force vector. ):\r\n<p style=\"text-align: center\">|<span style=\"text-decoration: underline\">M<\/span>| = +\/- |r\u22a5| |<span style=\"text-decoration: underline\">F<\/span>| [latex]\\hat{\\underline{k}}[\/latex]<\/p>\r\n* <em>though it's not always the [latex]\\hat{\\underline{k}}[\/latex] direction, it could be [latex]\\hat{\\underline{i}}[\/latex] or [latex]\\hat{\\underline{j}}[\/latex] as well. It depends on how you define your coordinate frame.<\/em>\r\n<h2>Example 4:<\/h2>\r\n<div><img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--1024x740.png\" alt=\"A step-by-step visual guide showing how to use the right-hand rule to determine the direction and magnitude of a moment using vector diagrams and hand photos.\" class=\"aligncenter wp-image-1168 size-large\" width=\"1024\" height=\"740\" \/><\/div>\r\n<div><\/div>\r\n<div>\r\n<h1><strong>3.1.3 Three-Finger Configuration<\/strong><\/h1>\r\nIf you find curling your fingers too confusing, you can try this method that uses your thumb, pointer finger, and middle finger all 90 degrees apart. Your<span style=\"text-decoration: underline\"> thumb is x<\/span>, your <span style=\"text-decoration: underline\">pointer finger is y<\/span>, your <span style=\"text-decoration: underline\">middle finger is z<\/span>.\r\n<div class=\"textbox\">\r\n<div class=\"page\" title=\"Page 840\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n<span>This is done by using your right hand, aligning your thumb with the first vector and your index with the second vector. The cross product will point in the direction of your middle finger (when you hold your middle finger perpendicular to the other two fingers). This is illustrated in the Figures below<\/span><span>. <\/span><img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-1024x694.png\" alt=\"A right-hand diagram illustrating the cross product of vectors vec a times vec b.\" class=\"aligncenter wp-image-1163\" width=\"453\" height=\"307\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_2693\" align=\"aligncenter\" width=\"300\"]<img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-16-161719-300x151.png\" alt=\"Two diagrams showing different orientations of a 3D coordinate system with x, y, and z axes.\" class=\"wp-image-2693 size-medium\" width=\"300\" height=\"151\" \/> Figure: Two possible orientations for a three-dimensional coordinate system. You can confirm using the right-hand rule that the z-axis is the cross product of x times y.[\/caption]\r\n\r\nSource: Introductory Physics, Ryan Martin et al.,<a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a>, pages 823\u2013825\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>The \"Curly Method\"<\/h2>\r\n<div>\r\n\r\nFor axial vectors, you use what I'm calling the curly method. To find whether the axis of rotation is positive or negative, curl your fingers in the direction of rotation and your thumb shows the direction of rotation, i.e. whether rotation is along the positive or negative x y or z direction. (This assumes you already have a coordinate frame defined to see which axis the wheel is rotating around and which direction).\r\n\r\nIf a wheel is rolling, the axis is what it rolls around. Curl your fingers in the direction of rotation and your thumb shows the direction of rotation.[footnote]Hand from page 127 of Calculus Based Physics, Jeffrey W. Schnick, <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7<\/a> &amp; tire from page 828 of Introductory Physics, Ryan Martin et al., <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a>, Edited by author.[\/footnote]\r\n\r\n<\/div>\r\n&nbsp;\r\n<p class=\"aligncenter wp-image-1170\"><img src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-1024x671.png\" alt=\"Diagrams showing the right-hand rule applied to wheel rotation, indicating positive and negative z-axis directions.\" class=\"aligncenter wp-image-1170\" width=\"610\" height=\"400\" \/><\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<em>Basically:<\/em> The right hand rule helps us to be consistent with how the x - y - z axes are oriented. It follows the rule that X x Y = Z. Using your fingers and thumb, there are two different methods. For one point, point your fingers in the direction of x, curl them towards y (you may have to flip your hand), and your thumb shows the direction of z. Trying to copy this 3d image onto your 2d page may be difficult, but with practice, you'll see the right angles between the drawn axes.\r\n\r\n<em>Application<\/em>: How do I know which way to push on the torque wrench to make the bolt on my wheel turn? If I point my thumb in the direction I want the bolt to move, and curl my fingers around the direction of the threads, I can see whether to push or pull on the wrench.\r\n\r\n<em>Looking Ahead:<\/em> We will calculate the moment many times throughout the rest of the book, and we need the right-hand rule every time, especially as we get into Chapter 4 and Rigid Body Equilibrium Equations.\r\n\r\n<\/div>\r\n<\/div>","rendered":"<p>Before we can analyze rigid bodies, we need to learn a little trick to help us with the cross product called the &#8216;right-hand rule&#8217;. We use the right-hand rule when we have two of the axes and need to find the direction of the third.<\/p>\n<div><\/div>\n<div>This is called a right-orthogonal system. The &#8216;<em>orthogonal&#8217;<\/em> part means that the three axes are all perpendicular to each other, and the&#8217; <em>right&#8217;<\/em> part means that [latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=\\underline{\\hat{k}}[\/latex], hence the right hand rule. Remember these from section 1.5?<\/div>\n<div>\n<ul>\n<li style=\"list-style-type: none\">\n<ul>\n<li style=\"list-style-type: none\">\n<ul>\n<li>[latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=\\underline{\\hat{k}}[\/latex]<\/li>\n<li>[latex]\\underline{\\hat{j}}\\times\\underline{\\hat{k}}=\\underline{\\hat{i}}[\/latex]<\/li>\n<li>[latex]\\underline{\\hat{k}}\\times\\underline{\\hat{i}}=\\underline{\\hat{j}}[\/latex]<\/li>\n<li>[latex]\\underline{\\hat{j}}\\times\\underline{\\hat{i}}=-\\underline{\\hat{k}}[\/latex]<\/li>\n<li>[latex]\\underline{\\hat{k}}\\times\\underline{\\hat{j}}=-\\underline{\\hat{i}}[\/latex]<\/li>\n<li>[latex]\\underline{\\hat{i}}\\times\\underline{\\hat{k}}=-\\underline{\\hat{j}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>The opposite of the right-orthogonal system is the left-orthogonal system where [latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=-\\underline{\\hat{k}}[\/latex]. We don&#8217;t use that one!<\/p>\n<p>There are two ways to do the right hand rule, and they take practice to conceptually understand, but this will make solving problems much quicker. You&#8217;re going to use your fingers and thumb to represent the x, y, and z axes.<\/p>\n<h1><strong>3.1.1 The Whole-Hand Method<\/strong><\/h1>\n<div>In the following description, A x B = C, so for the coordinate frame, X x Y = Z ([latex]\\underline{\\hat{i}}\\times\\underline{\\hat{j}}=\\underline{\\hat{k}}[\/latex]). Your fingers go in the direction of X, then you\u00a0 bend them 90 degrees to point towards Y, and your thumb is in the direction of Z.<\/div>\n<div><\/div>\n<div class=\"textbox\">\n<p>The direction of the cross product vector A x B is given by the right-hand rule for the cross product of two vectors. To apply this right-hand rule, extend the fingers of your right hand so that they are pointing directly away from your right elbow. Extend your thumb so that it is at<br \/>\nright angles to your fingers.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0519.jpg\" alt=\"A drawing of a right hand with the thumb extended upward and fingers pointing forward.\" class=\"aligncenter wp-image-257\" width=\"253\" height=\"208\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0519.jpg 505w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0519-300x247.jpg 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0519-65x54.jpg 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0519-225x185.jpg 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0519-350x288.jpg 350w\" sizes=\"auto, (max-width: 253px) 100vw, 253px\" \/><\/p>\n<p>Keeping your fingers aligned with your forearm, point your fingers in the direction of the first vector (the one that appears before the \u201c\u00d7\u201d in the mathematical expression for the cross product; e.g. the A in A x B ).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520.jpg\" alt=\"A right-hand diagram showing the thumb pointing up (B) and fingers pointing right (A).\" class=\"aligncenter wp-image-258\" width=\"293\" height=\"254\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520.jpg 1115w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520-300x259.jpg 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520-1024x885.jpg 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520-768x664.jpg 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520-65x56.jpg 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520-225x195.jpg 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0520-350x303.jpg 350w\" sizes=\"auto, (max-width: 293px) 100vw, 293px\" \/><\/p>\n<p>Now rotate your hand, as necessary, about an imaginary axis extending along your forearm and along your middle finger, until your hand is oriented such that, if you were to close your fingers, they would point in the\u00a0direction of the second vector.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521.jpg\" alt=\"A right-hand diagram with the thumb pointing out of the page, fingers pointing right (A), and palm facing upward with vector B.\" class=\"aligncenter wp-image-259\" width=\"326\" height=\"190\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521.jpg 1477w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521-300x176.jpg 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521-1024x600.jpg 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521-768x450.jpg 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521-65x38.jpg 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521-225x132.jpg 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/IMG_0521-350x205.jpg 350w\" sizes=\"auto, (max-width: 326px) 100vw, 326px\" \/><\/p>\n<p>Your thumb is now pointing in the direction of the cross product vector. C = A x B. The cross product vector C is always perpendicular to both of the vectors that are in the cross product (the A and the B in the case at hand). Hence, if you draw them so that both of the vectors that are in the cross product are in the plane of the page, the cross product vector will always be perpendicular to the page, either straight into the page, or straight out of the page. In the case at hand, it is straight out of the page.<\/p>\n<p>When we use the cross product to calculate the torque due to a force F whose point of application has a position vector r, relative to the point about which we are calculating the torque, we get an axial torque vector \u03c4. To determine the sense of rotation that such a torque vector would correspond to, about the axis defined by the torque vector itself, we use the Right Hand Rule for Something Curly Something Straight. Note that we are calculating the torque with respect to a point rather than an axis\u2014the axis about which the torque acts comes out in the answer.<\/p>\n<p>&nbsp;<\/p>\n<p>Source: <span>Jeffrey W. Schnick <\/span><a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7<\/a> pages 135\u2013137<\/p>\n<\/div>\n<p>The hardest part of the right-hand rule is imagining the different axes and envisioning how they are perpendicular to each other. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-1024x218.png\" alt=\"Diagrams showing the x, y, and z axes with right angle symbols to illustrate perpendicularity between each axis pair.\" class=\"aligncenter wp-image-1126 size-large\" width=\"1024\" height=\"218\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-1024x218.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-300x64.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-768x163.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-1536x326.png 1536w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-65x14.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-225x48.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM-350x74.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.00.21-PM.png 2014w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>Try this one in 2d and 3d. Imagine (or draw) the right-angle symbols (Answer will be in a few steps)<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM.png\" alt=\"Two diagrams showing right-handed coordinate systems with x, y, and z axes.\" class=\"aligncenter wp-image-1127\" width=\"306\" height=\"110\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM.png 1012w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM-300x108.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM-768x276.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM-65x23.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM-225x81.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.03.47-PM-350x126.png 350w\" sizes=\"auto, (max-width: 306px) 100vw, 306px\" \/><\/p>\n<h2><strong>Example 1:<\/strong><\/h2>\n<p>Using these x and y, let&#8217;s use the right-hand rule to find the direction of z.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-1024x304.png\" alt=\"Two 3D axis diagrams showing right-angle symbols between x, y, and z axes in different colors.\" class=\"aligncenter wp-image-1130\" width=\"579\" height=\"172\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-1024x304.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-300x89.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-768x228.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-65x19.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-225x67.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1-350x104.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.04.23-PM-1.png 1158w\" sizes=\"auto, (max-width: 579px) 100vw, 579px\" \/><\/p>\n<p>Here are the steps you can follow:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-1024x748.png\" alt=\"A 4-step visual guide demonstrating the right-hand rule for determining 3D coordinate directions using photos of a hand and labeled axes.\" class=\"aligncenter wp-image-1131 size-large\" width=\"1024\" height=\"748\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-1024x748.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-300x219.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-768x561.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-1536x1122.png 1536w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-65x47.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-225x164.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM-350x256.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-12.59.54-PM.png 1862w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3>Example 2:<\/h3>\n<p>Sometimes you will need to flip your hand 180 degrees to find which way lets you point your fingers in the y direction, for example:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-1024x1005.png\" alt=\"A step-by-step guide using hand photos and 3D coordinate diagrams to explain the right-hand rule and identify the correct direction of the z-axis.\" class=\"aligncenter wp-image-1150 size-large\" width=\"1024\" height=\"1005\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-1024x1005.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-300x295.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-768x754.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-65x64.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-225x221.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM-350x344.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.16.55-PM.png 1436w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3>Example 3:<\/h3>\n<p>It&#8217;s important for you to be able to envision how the axes are perpendicular. Now practice using the right hand rule if you are trying to find x.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-1024x263.png\" alt=\"A visual explanation of using the right-hand rule to find the x-axis direction when z and y are known, showing hand positioning and a 3D coordinate system.\" class=\"aligncenter wp-image-1151 size-large\" width=\"1024\" height=\"263\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-1024x263.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-300x77.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-768x197.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-1536x395.png 1536w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-65x17.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-225x58.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM-350x90.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.21.07-PM.png 1610w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h3>Your Turn!<\/h3>\n<p>Keep going with these examples. The rules stay the same: thumb towards z, curled fingers towards y, extended fingers towards x. Find the missing axis:<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-1024x207.png\" alt=\"Five differently colored 3D coordinate axes showing various orientations of the x, y, and z directions.\" class=\"aligncenter wp-image-1152 size-large\" width=\"1024\" height=\"207\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-1024x207.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-300x61.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-768x155.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-1536x311.png 1536w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-65x13.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-225x46.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM-350x71.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.25.56-PM.png 1928w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p style=\"text-align: center\">.<\/p>\n<p style=\"text-align: center\">.<\/p>\n<p style=\"text-align: center\">Did you do it?<\/p>\n<p style=\"text-align: center\">.<\/p>\n<p style=\"text-align: center\">.<\/p>\n<p style=\"text-align: center\">.<\/p>\n<p style=\"text-align: center\">Here are the answers:<\/p>\n<p style=\"text-align: center\">.<\/p>\n<p style=\"text-align: center\">.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-1024x248.png\" alt=\"Five colored 3D coordinate systems with green arrows indicating the correct direction of the third axis using the right-hand rule.\" class=\"aligncenter wp-image-1153 size-large\" width=\"1024\" height=\"248\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-1024x248.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-300x73.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-768x186.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-1536x372.png 1536w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-65x16.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-225x54.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM-350x85.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.27.21-PM.png 1974w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h1>3.1.2 Right Hand Rule and Torque<\/h1>\n<p>The third way to calculate torque, as was alluded to in Section 1.6, is to use the right-hand rule to identify the axis of rotation. The first way (the scalar method) uses | <span><span style=\"text-decoration: underline\"><strong>M<\/strong><\/span> |<\/span> = |<span style=\"text-decoration: underline\"><strong>r<\/strong><\/span>| |<span style=\"text-decoration: underline\"><strong>F<\/strong><\/span>| sin \u0398, and often the angle between the position vector and force is 90 degrees. The vector method is for more complicated situations and uses the cross product <span style=\"text-decoration: underline\"> <strong>r<\/strong><\/span> x <span style=\"text-decoration: underline\"><strong>F<\/strong><\/span><strong> = <\/strong><span style=\"text-decoration: underline\"><strong>M<\/strong><\/span><strong>. <\/strong>The third method finds the scalar value separately, then uses the right-hand rule to find the direction (positive or negative along the third axis).<\/p>\n<ul>\n<li>Point your fingers in the direction of <em><span style=\"text-decoration: underline\">the perpendicular part<\/span><\/em> of the position vector <span style=\"text-decoration: underline\"><strong>r<span><\/span><\/strong><\/span> (as you would for x)<\/li>\n<li>Curl them towards the direction of the Force vector <span style=\"text-decoration: underline\"><strong>F<\/strong><\/span> (as you would for y)<\/li>\n<li>Your thumb is in the direction of the moment <span style=\"text-decoration: underline\"><strong>M<\/strong><\/span> that results from the force (as for z)<\/li>\n<\/ul>\n<p>The following will help you understand what is meant by: <em>the perpendicular part <\/em>of the position vector:<\/p>\n<div class=\"textbox\">\n<div class=\"page\" title=\"Page 132\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><span>The torque <\/span><span>\u03c4 <\/span><span>can be expressed as the cross product of the position vector <\/span><span>r <\/span><span>for the point of application of the force, and the force vector <\/span><span style=\"text-decoration: underline\"><strong>F<\/strong><\/span> <span>itself: <span style=\"text-decoration: underline\"><strong>r<\/strong><\/span> x <span style=\"text-decoration: underline\"><strong>F<\/strong><\/span><strong> =\u00a0<\/strong><span style=\"text-decoration: underline\"><strong>M<\/strong><\/span><\/span><\/p>\n<p><span>Before we begin our mathematical discussion of what we mean by the cross product, a few words about the vector <strong>r <\/strong>are in order. It is important for you to be able to distinguish between the position vector <strong>r <\/strong>for the force, and the moment arm, so we present them below in one and the same diagram. We use the same example that we have used before: <span style=\"text-decoration: underline\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-1024x579.png\" alt=\"An irregular shaped object with a labeled axis of rotation at point O and a force F applied at a distant point.\" class=\"aligncenter wp-image-1155\" width=\"436\" height=\"247\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-1024x579.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-300x170.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-768x434.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-65x37.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-225x127.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM-350x198.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.56.17-PM.png 1128w\" sizes=\"auto, (max-width: 436px) 100vw, 436px\" \/><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><span>in which we are looking directly along the axis of rotation (so it looks like a dot) and the force lies in a plane perpendicular to that axis of rotation. We use the diagramatic convention that, the point at which the force is applied to the rigid body is the point at which one end of the arrow in <\/span><span>the diagram touches the rigid body. Now we add the line of action of the force and the moment arm r\u22a5 to the diagram, as well as the position vector r of the point of application of the force. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-1024x685.png\" alt=\"A diagram showing an object with force vec F, position vector vec r, moment arm r_perpendicular, and the line of action of the force labeled.\" class=\"aligncenter wp-image-1156\" width=\"441\" height=\"295\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-1024x685.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-300x201.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-768x514.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-65x43.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-225x151.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM-350x234.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.58.23-PM.png 1190w\" sizes=\"auto, (max-width: 441px) 100vw, 441px\" \/><\/span><span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"page\" title=\"Page 133\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><span>The moment arm can actually be defined in terms of the position vector for the point of application of the force. Consider a tilted <\/span><span>x<\/span><span>-y coordinate system, having an origin on the axis of rotation, with one axis parallel to the line of action of the force and one axis perpendicular to the line of action of the force. We label the <\/span><span>x <\/span><span>axis <\/span><span>\u2534 <\/span><span>for \u201cperpendicular\u201d and the y axis <\/span><span>|| <\/span><span>for &#8220;parallel&#8221;.<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM.png\" alt=\"Diagram showing a force vector vec F, position vector vec r, and red dashed lines representing parallel and perpendicular directions from the axis of rotation.\" class=\"aligncenter wp-image-1157\" width=\"354\" height=\"360\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM.png 976w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM-295x300.png 295w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM-768x781.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM-65x66.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM-225x229.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-2.59.06-PM-350x356.png 350w\" sizes=\"auto, (max-width: 354px) 100vw, 354px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"page\" title=\"Page 134\">\n<div class=\"section\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><span>Now we break up the position vector <\/span><span>r <\/span><span>into its component vectors along the \u2534 (perpendicular) and || (parallel) axes. <\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM.png\" alt=\"Diagram showing force vec F, position vector vec r, and its components r_perpendicular and r_parallel with dashed red lines indicating perpendicular and parallel directions.\" class=\"aligncenter wp-image-1158\" width=\"343\" height=\"284\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM.png 994w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM-300x248.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM-768x635.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM-65x54.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM-225x186.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-3.00.33-PM-350x289.png 350w\" sizes=\"auto, (max-width: 343px) 100vw, 343px\" \/><\/p>\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><span>From the diagram it is clear that the moment arm <\/span><span>r <\/span><span>is just the magnitude of the component <\/span><span>\u2534 <\/span><span>vector, in the perpendicular-to-the-force direction, of the position vector of the point of application of the force. <\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"layoutArea\">\n<div class=\"column\"><\/div>\n<\/div>\n<div>Source: <span>Calculus\u00a0Based Physics, Jeffrey W. Schnick,\u00a0<\/span><a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7\" class=\"\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7<\/a> pages 132\u2013137<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>You use the right-hand rule twice during this method to find the vector. First, to determine the coordinate frame and again to see in which direction the torque is aligned. Then you multiply by the magnitude of the perpendicular portion of the position vector (r\u22a5 or the &#8220;moment arm&#8221;) and the magnitude of the force vector. ):<\/p>\n<p style=\"text-align: center\">|<span style=\"text-decoration: underline\">M<\/span>| = +\/- |r\u22a5| |<span style=\"text-decoration: underline\">F<\/span>| [latex]\\hat{\\underline{k}}[\/latex]<\/p>\n<p>* <em>though it&#8217;s not always the [latex]\\hat{\\underline{k}}[\/latex] direction, it could be [latex]\\hat{\\underline{i}}[\/latex] or [latex]\\hat{\\underline{j}}[\/latex] as well. It depends on how you define your coordinate frame.<\/em><\/p>\n<h2>Example 4:<\/h2>\n<div><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--1024x740.png\" alt=\"A step-by-step visual guide showing how to use the right-hand rule to determine the direction and magnitude of a moment using vector diagrams and hand photos.\" class=\"aligncenter wp-image-1168 size-large\" width=\"1024\" height=\"740\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--1024x740.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--300x217.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--768x555.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--65x47.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--225x163.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot--350x253.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/screenshot-.png 1502w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/div>\n<div><\/div>\n<div>\n<h1><strong>3.1.3 Three-Finger Configuration<\/strong><\/h1>\n<p>If you find curling your fingers too confusing, you can try this method that uses your thumb, pointer finger, and middle finger all 90 degrees apart. Your<span style=\"text-decoration: underline\"> thumb is x<\/span>, your <span style=\"text-decoration: underline\">pointer finger is y<\/span>, your <span style=\"text-decoration: underline\">middle finger is z<\/span>.<\/p>\n<div class=\"textbox\">\n<div class=\"page\" title=\"Page 840\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><span>This is done by using your right hand, aligning your thumb with the first vector and your index with the second vector. The cross product will point in the direction of your middle finger (when you hold your middle finger perpendicular to the other two fingers). This is illustrated in the Figures below<\/span><span>. <\/span><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-1024x694.png\" alt=\"A right-hand diagram illustrating the cross product of vectors vec a times vec b.\" class=\"aligncenter wp-image-1163\" width=\"453\" height=\"307\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-1024x694.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-300x203.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-768x521.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-65x44.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-225x153.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM-350x237.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-5.06.16-PM.png 1198w\" sizes=\"auto, (max-width: 453px) 100vw, 453px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_2693\" aria-describedby=\"caption-attachment-2693\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-16-161719-300x151.png\" alt=\"Two diagrams showing different orientations of a 3D coordinate system with x, y, and z axes.\" class=\"wp-image-2693 size-medium\" width=\"300\" height=\"151\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-16-161719-300x151.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-16-161719-65x33.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-16-161719-225x113.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-16-161719-350x176.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screenshot-2025-07-16-161719.png 502w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-2693\" class=\"wp-caption-text\">Figure: Two possible orientations for a three-dimensional coordinate system. You can confirm using the right-hand rule that the z-axis is the cross product of x times y.<\/figcaption><\/figure>\n<p>Source: Introductory Physics, Ryan Martin et al.,<a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a>, pages 823\u2013825<\/p>\n<\/div>\n<\/div>\n<h2>The &#8220;Curly Method&#8221;<\/h2>\n<div>\n<p>For axial vectors, you use what I&#8217;m calling the curly method. To find whether the axis of rotation is positive or negative, curl your fingers in the direction of rotation and your thumb shows the direction of rotation, i.e. whether rotation is along the positive or negative x y or z direction. (This assumes you already have a coordinate frame defined to see which axis the wheel is rotating around and which direction).<\/p>\n<p>If a wheel is rolling, the axis is what it rolls around. Curl your fingers in the direction of rotation and your thumb shows the direction of rotation.<a class=\"footnote\" title=\"Hand from page 127 of Calculus Based Physics, Jeffrey W. Schnick, https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7 &amp; tire from page 828 of Introductory Physics, Ryan Martin et al., https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb, Edited by author.\" id=\"return-footnote-162-1\" href=\"#footnote-162-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p class=\"aligncenter wp-image-1170\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-1024x671.png\" alt=\"Diagrams showing the right-hand rule applied to wheel rotation, indicating positive and negative z-axis directions.\" class=\"aligncenter wp-image-1170\" width=\"610\" height=\"400\" srcset=\"https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-1024x671.png 1024w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-300x197.png 300w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-768x503.png 768w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-65x43.png 65w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-225x147.png 225w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM-350x229.png 350w, https:\/\/pressbooks.library.upei.ca\/statics\/wp-content\/uploads\/sites\/56\/2021\/03\/Screen-Shot-2021-07-30-at-9.37.47-PM.png 1474w\" sizes=\"auto, (max-width: 610px) 100vw, 610px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><em>Basically:<\/em> The right hand rule helps us to be consistent with how the x &#8211; y &#8211; z axes are oriented. It follows the rule that X x Y = Z. Using your fingers and thumb, there are two different methods. For one point, point your fingers in the direction of x, curl them towards y (you may have to flip your hand), and your thumb shows the direction of z. Trying to copy this 3d image onto your 2d page may be difficult, but with practice, you&#8217;ll see the right angles between the drawn axes.<\/p>\n<p><em>Application<\/em>: How do I know which way to push on the torque wrench to make the bolt on my wheel turn? If I point my thumb in the direction I want the bolt to move, and curl my fingers around the direction of the threads, I can see whether to push or pull on the wrench.<\/p>\n<p><em>Looking Ahead:<\/em> We will calculate the moment many times throughout the rest of the book, and we need the right-hand rule every time, especially as we get into Chapter 4 and Rigid Body Equilibrium Equations.<\/p>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-162-1\">Hand from page 127 of Calculus Based Physics, Jeffrey W. Schnick, <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=ce74a181-ccde-491c-848d-05489ed182e7<\/a> &amp; tire from page 828 of Introductory Physics, Ryan Martin et al., <a href=\"https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb\">https:\/\/openlibrary.ecampusontario.ca\/catalogue\/item\/?id=4c3c2c75-0029-4c9e-967f-41f178bebbbb<\/a>, Edited by author. <a href=\"#return-footnote-162-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":74,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-162","chapter","type-chapter","status-publish","hentry"],"part":54,"_links":{"self":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/162","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/users\/74"}],"version-history":[{"count":29,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions"}],"predecessor-version":[{"id":2795,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions\/2795"}],"part":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapters\/162\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/media?parent=162"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/pressbooks\/v2\/chapter-type?post=162"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/contributor?post=162"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/statics\/wp-json\/wp\/v2\/license?post=162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}