1.4 Dot Product

A vector can be multiplied by another vector but may not be divided by another vector. There are two kinds of products of vectors used broadly in physics and engineering. One kind of multiplication is a scalar multiplication of two vectors. Taking a scalar product of two vectors results in a number (a scalar), as its name indicates. Scalar products are used to define work and energy relations. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector. A quite different kind of multiplication is a vector multiplication of vectors. Taking a vector product of two vectors returns as a result a vector, as its name suggests. Vector products are used to define other derived vector quantities. For example, in describing rotations, a vector quantity called torque is defined as a vector product of an applied force (a vector) and its distance from pivot to force (a vector). It is important to distinguish between these two kinds of vector multiplications because the scalar product is a scalar quantity and a vector product is a vector quantity.

Scalar multiplication of two vectors yields a scalar product.

When the vectors are given in their vector component forms:

$$\vec A=A_x\underline{\hat{i}}+A_y\underline{\hat{j}}+A_z\underline{\hat{k}}\text{  and  }\vec B=B_x\underline{\hat{i}}+B_y\underline{\hat{j}}+B_z\underline{\hat{k}}$$

we can compute their scalar product as follows:

$$\vec A\cdot\vec B=(A_x\underline{\hat{i}}+A_y\underline{\hat{j}}+A_z\underline{\hat{k}})\cdot(B_x\underline{\hat{i}}+B_y\underline{\hat{j}}+B_z\underline{\hat{k}})\\=A_xB_x\underline{\hat{i}}\cdot\underline{\hat{i}}+A_xB_y\underline{\hat{i}}\cdot\underline{\hat{j}}+A_xB_z\underline{\hat{i}}\cdot\underline{\hat{k}}\\+A_yB_x\underline{\hat{j}}\cdot\underline{\hat{i}}+A_yB_y\underline{\hat{j}}\cdot\underline{\hat{j}}+A_yB_z\underline{\hat{j}}\cdot\underline{\hat{k}}\\+A_zB_x\underline{\hat{k}}\cdot\underline{\hat{i}}+A_zB_y\underline{\hat{k}}\cdot\underline{\hat{j}}+A_zB_z\underline{\hat{k}}\cdot\underline{\hat{k}}$$

Since scalar products of two different unit vectors of axes give zero, and scalar products of unit vectors with themselves give one, there are only three nonzero terms in this expression. Thus, the scalar product simplifies to:

[latex]\vec A\cdot\vec B=A_xB_x+A_yB_y+A_zB_z[/latex]

We can use the equation below to find the angle between two vectors. When we divide [latex]\vec A\cdot\vec B=|\vec A||\vec B| \cos\phi[/latex] by [latex]|\vec A || \vec B|[/latex] , we obtain the equation for cos(ϕ), into which we substitute the equation from above:

$$\cos\phi=\frac{\vec A\cdot\vec B}{|\vec A||\vec B| }=\frac{A_xB_x+A_yB_y+A_zB_z}{|\vec A||\vec B| }$$

Angle $\text{ϕ}$ between vectors [latex]\vec A \text{ and }\vec B[/latex] is obtained by taking the inverse cosine of the expression above. 

Source: University Physics Volume 1, OpenStax CNX, https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/2-4-products-of-vectors (many examples at this page).