# 2.3 The dot product

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• Calculate the dot product of two given vectors.
• Determine whether two given vectors are perpendicular.
• Find the direction cosines of a given vector.
• Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it.
• Calculate the work done by a given force.

If we apply a force to an object so that the object moves, we say that work is done by the force. In Introduction to Applications of Integration on integration applications, we looked at a constant force and we assumed the force was applied in the direction of motion of the object. Under those conditions, work can be expressed as the product of the force acting on an object and the distance the object moves. In this chapter, however, we have seen that both force and the motion of an object can be represented by vectors.

In this section, we develop an operation called the dot product , which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. It even provides a simple test to determine whether two vectors meet at a right angle.

## The dot product and its properties

We have already learned how to add and subtract vectors. In this chapter, we investigate two types of vector multiplication. The first type of vector multiplication is called the dot product, based on the notation we use for it, and it is defined as follows:

## Definition

The dot product of vectors $\text{u}=⟨{u}_{1},{u}_{2},{u}_{3}⟩$ and $\text{v}=⟨{v}_{1},{v}_{2},{v}_{3}⟩$ is given by the sum of the products of the components

$\text{u}·\text{v}={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+{u}_{3}{v}_{3}.$

Note that if u and v are two-dimensional vectors, we calculate the dot product in a similar fashion. Thus, if $\text{u}=⟨{u}_{1},{u}_{2}⟩$ and $\text{v}=⟨{v}_{1},{v}_{2}⟩,$ then

$\text{u}·\text{v}={u}_{1}{v}_{1}+{u}_{2}{v}_{2}.$

When two vectors are combined under addition or subtraction, the result is a vector. When two vectors are combined using the dot product, the result is a scalar. For this reason, the dot product is often called the scalar product . It may also be called the inner product .

## Calculating dot products

1. Find the dot product of $\text{u}=⟨3,5,2⟩$ and $\text{v}=⟨-1,3,0⟩.$
2. Find the scalar product of $\text{p}=10\text{i}-4\text{j}+7\text{k}$ and $\text{q}=-2\text{i}+\text{j}+6\text{k}.$
1. Substitute the vector components into the formula for the dot product:
$\begin{array}{cc}\hfill \text{u}·\text{v}& ={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+{u}_{3}{v}_{3}\hfill \\ & =3\left(-1\right)+5\left(3\right)+2\left(0\right)=-3+15+0=12.\hfill \end{array}$
2. The calculation is the same if the vectors are written using standard unit vectors. We still have three components for each vector to substitute into the formula for the dot product:
$\begin{array}{cc}\hfill \text{p}·\text{q}& ={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+{u}_{3}{v}_{3}\hfill \\ & =10\left(-2\right)+\left(-4\right)\left(1\right)+\left(7\right)\left(6\right)=-20-4+42=18.\hfill \end{array}$

Find $\text{u}·\text{v},$ where $\text{u}=⟨2,9,-1⟩$ and $\text{v}=⟨-3,1,-4⟩.$

7

Like vector addition and subtraction, the dot product has several algebraic properties. We prove three of these properties and leave the rest as exercises.

## Properties of the dot product

Let $\text{u},$ $\text{v},$ and $\text{w}$ be vectors, and let c be a scalar.

$\begin{array}{ccccccccc}\text{i.}\hfill & & & \hfill \text{u}·\text{v}& =\hfill & \text{v}·\text{u}\hfill & & & \text{Commutative property}\hfill \\ \text{ii.}\hfill & & & \hfill \text{u}·\left(\text{v}+\text{w}\right)& =\hfill & \text{u}·\text{v}+\text{u}·\text{w}\hfill & & & \text{Distributive property}\hfill \\ \text{iii.}\hfill & & & \hfill c\left(\text{u}·\text{v}\right)& =\hfill & \left(c\text{u}\right)·\text{v}=\text{u}·\left(c\text{v}\right)\hfill & & & \text{Associative property}\hfill \\ \text{iv.}\hfill & & & \hfill \text{v}·\text{v}& =\hfill & {‖\text{v}‖}^{2}\hfill & & & \text{Property of magnitude}\hfill \end{array}$

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