Section 9.3The Dot Product

In the last section, we considered vector addition and scalar multiplication and found that each operation had a natural geometric interpretation. In this section, we will introduce a means of multiplying vectors.

The Dot Product

The definition of the dot product for vectors in \(\R^2\) given in Preview Activity 9.3.1 can be extended to vectors in \(\R^n\).

For instance, we find that \[ \langle 3, 0, 1 \rangle\cdot\langle -2, 1, 4\rangle = 3\cdot(-2) + 0\cdot1 + 1\cdot4 = -6 + 0 + 4 = -2. \]

Notice that the resulting quantity is a scalar. Our work in Preview Activity 9.3.1 examined dot products of two-dimensional vectors.

The dot product is a natural way to define a product of two vectors. In addition, it behaves in ways that are similar to the product of, say, real numbers.

Moreover, the dot product gives us valuable geometric information about the vectors and their relative orientation. For instance, let's consider what happens when we dot a vector with itself: \[ \vu\cdot\vu = \langle u_1,u_2,\ldots,u_n \rangle \cdot \langle u_1,u_2,\ldots,u_n \rangle = u_1^2 + u_2^2 + \ldots + u_n^2 = |\vu|^2. \]

In other words, the dot product of a vector with itself gives the square of the length of the vector: \(\vu\cdot\vu=|\vu|^2\).

The angle between vectors

The dot product can help us understand the angle between two vectors. For instance, if we are given two vectors \(\vu\) and \(\vv\), there are two angles that these vectors create, as depicted at left in Figure 9.3.4. We will call \(\theta\), the smaller of these angles, the angle between these vectors. Notice that \(\theta\) lies between 0 and \(\pi\).

To determine this angle, we may apply the Law of Cosines to the triangle shown at right in Figure 9.3.4.

Using the fact that the dot product of a vector with itself gives us the square of its length, together with the properties of the dot product, we find: \[\begin{aligned}|\vu-\vv|^2 =\mathstrut & |\vu|^2 + |\vv|^2 - 2|\vu||\vv|\cos(\theta) \\ (\vu-\vv)\cdot(\vu-\vv) =\mathstrut & \vu\cdot\vu + \vv\cdot\vv - 2|\vu||\vv|\cos(\theta) \\ \vu\cdot(\vu-\vv) - \vv\cdot(\vu-\vv)=\mathstrut & \vu\cdot\vu + \vv\cdot\vv - 2|\vu||\vv|\cos(\theta) \\ \vu\cdot\vu - 2\vu\cdot\vv + \vv\cdot\vv =\mathstrut & \vu\cdot\vu + \vv\cdot\vv - 2|\vu||\vv|\cos(\theta) \\ -2\vu\cdot\vv =\mathstrut & -2|\vu||\vv|\cos(\theta) \\ \vu\cdot\vv =\mathstrut & |\vu||\vv|\cos(\theta).\end{aligned}\]

To summarize, we have the important relationship \[ \vu\cdot\vv = u_1v_1+u_2v_2 + \ldots + u_nv_n = |\vu||\vv|\cos(\theta). \]

It is sometimes useful to think of Equation  as giving us an expression for the angle between two vectors: \[ \theta = \cos^{-1}\left(\frac{\vu\cdot\vv}{|\vu||\vv|}\right). \]

The real beauty of this expression is this: the dot product is a very simple algebraic operation to perform yet it provides us with important geometric information — namely the angle between the vectors — that would be difficult to determine otherwise.

The Dot Product and Orthogonality

When the angle between two vectors is a right angle, it is frequently the case that something important is happening. In this case, we say the vectors are orthogonal. For instance, orthogonality often plays a role in optimization problems; to determine the shortest path from a point in \(\R^3\) to a given plane, we move along a line orthogonal to the plane.

As Activity 9.3.3 indicates, the dot product provides a simple means to determine whether two vectors are orthogonal to one another. In this case, \(\vu\cdot\vv=|\vu||\vv|\cos(\pi/2) = 0\), so we make the following important observation.

More generally, the sign of the dot product gives us useful information about the relative orientation of the vectors. If we remember that \[\begin{aligned}\cos(\theta) \gt 0 \mathstrut & \ \ \ \text{ if } \theta \text{ is an acute angle,} \\ \cos(\theta) = 0 \mathstrut & \ \ \ \text{ if } \theta \text{ is a right angle, and} \\ \cos(\theta) \lt 0 \mathstrut & \ \ \ \text{ if } \theta \text{ is an obtuse angle,}\end{aligned}\] we see that for nonzero vectors \(\vu\) and \(\vv\), \[\begin{aligned} \vu\cdot\vv \gt 0 \mathstrut & \ \ \ \text{ if } \theta \text{ is an acute angle,} \\ \vu\cdot\vv = 0 \mathstrut & \ \ \ \text{ if } \theta \text{ is a right angle, and} \\ \vu\cdot\vv \lt 0 \mathstrut & \ \ \ \text{ if } \theta \text{ is an obtuse angle.}\end{aligned}\] This is illustrated in Figure 9.3.5.

Work, Force, and Displacement

In physics, work is a measure of the energy required to apply a force to an object through a displacement. For instance, Figure 9.3.6 shows a force \(\vF\) displacing an object from point \(A\) to point \(B\). The displacement is then represented by the vector \(\overrightarrow{AB}\).

It turns out that the work required to displace the object is \[ W = \vF\cdot\overrightarrow{AB} = |\vF||\overrightarrow{AB}|\cos(\theta). \]

This means that the work is determined only by the magnitude of the force applied parallel to the displacement. Consequently, if we are given two vectors \(\vu\) and \(\vv\), we would like to write \(\vu\) as a sum of two vectors, one of which is parallel to \(\vv\) and one of which is orthogonal to \(\vv\). We take up this task after the next activity.

Projections

Suppose we are given two vectors \(\vu\) and \(\vv\) as shown at left in Figure 9.3.7. Motivated by our discussion of work, we would like to write \(\vu\) as a sum of two vectors, one of which is parallel to \(\vv\) and one of which is orthogonal. That is, we would like to write \[ \vu = \proj_{\vv}\vu + \proj_{\perp\vv}\vu, \] where \(\proj_{\vv}\vu\) is parallel to \(\vv\) and \(\proj_{\perp\vv}\vu\) is orthogonal to \(\vv\). We call the vector \(\proj_{\vv}\vu\) the projection of \(\vu\) onto \(\vv\). Note that, as the diagram at right in Figure 9.3.7 illustrates, it is also possible to create a projection even if the angle between the vectors \(\vu\) and \(\vv\) exceeds \(\frac\pi2\).

To find the vector \(\proj_{\vv} \vu\), we will dot both sides of Equation  with the vector \(\vv\), to find that \[\begin{aligned}\vu\cdot\vv=\mathstrut & (\proj_{\vv}\vu + \proj_{\perp\vv}\vu)\cdot\vv \\ =\mathstrut & (\proj_{\vv}\vu)\cdot\vv + (\proj_{\perp\vv}\vu)\cdot\vv \\ =\mathstrut & (\proj_{\vv}\vu)\cdot\vv.\end{aligned}\]

Notice that \((\proj_{\perp\vv}\vu)\cdot\vv = 0\) since \(\proj_{\perp\vv}\vu\) is orthogonal to \(\vv\). Also, \(\proj_{\vv}\vu\) must be a scalar multiple of \(\vv\) since it is parallel to \(\vv\), so we will write \(\proj_{\vv}\vu = s\vv\). It follows that \[ \vu\cdot\vv =(\proj_{\vv}\vu)\cdot\vv = s\vv\cdot\vv, \] which means that \[ s = \frac{\vu\cdot\vv}{\vv\cdot\vv} \] and hence \[ \proj_{\vv}\vu = \frac{\vu\cdot\vv}{\vv\cdot\vv}\vv = \frac{\vu\cdot\vv}{|\vv|^2}\vv \]

It is sometimes useful to write \(\proj_{\vv}\vu\) as a scalar times a unit vector in the direction of \(\vv\). We call this scalar the component of \(\vu\) along \(\vv\) and denote it as \(\comp_{\vv}\vu\). We therefore have \[ \proj_{\vv}\vu = \frac{\vu\cdot\vv}{|\vv|^2}\vv = \frac{\vu\cdot\vv}{|\vv|} \frac{\vv}{|\vv|}= \comp_{\vv}\vu \frac{\vv}{|\vv|}, \] so that \[ \comp_{\vv}\vu = \frac{\vu\cdot\vv}{|\vv|}. \]

Moreover, since \[ \vu = \proj_{\vv} \vu + \proj_{\perp \vv} \vu, \] it follows that \[ \proj_{\perp \vv} \vu = \vu - \proj_{\vv} \vu. \]

This shows that once we have computed \(\proj_{\vv} \vu\), we can find \(\proj_{\perp \vv} \vu\) simply by calculating the difference of two known vectors.

Summary

• The dot product of two vectors in \(\R^n\), \(\vu = \langle u_1, u_2, \ldots, u_n \rangle\) and \(\vv = \langle v_1, v_2, \ldots, v_n \rangle\), is the scalar \[ \vu \cdot \vv = u_1v_1 + u_2v_2 + \cdots + u_nv_n. \]

• The dot product is related to the length of a vector since \(\vu\cdot\vu = |\vu|^2\).

• The dot product provides us with information about the angle between the vectors since \[ \vu\cdot\vv = |\vu| \ |\vv|\cos(\theta), \] where \(\theta\) is the angle between \(\vu\) and \(\vv\).

• Two vectors are orthogonal if the angle between them is \(\pi/2\). In terms of the dot product, the vectors \(\vu\) and \(\vv\) are orthogonal if and only if \(\vu \cdot \vv = 0\).

• The projection of a vector \(\vu\) in \(\R^n\) onto a vector \(\vv\) in \(\R^n\) is the vector \[ \proj_{\vv} \vu = \frac{\vu \cdot \vv}{\vv\cdot\vv} \vv. \]

Exercises

1

Find \(\mathbf a \cdot \mathbf b\) if

\(\mathbf a = \langle -4, 3, -4 \rangle\) and \(\mathbf b = \langle 4, 4, -1 \rangle\)

\(\mathbf a \cdot \mathbf b =\)____________________

Is the angle between the vectors "acute", "obtuse" or "right"?

____________________

Show answer

\(0\), \(right\)

2

Determine if the pairs of vectors below are "parallel", "orthogonal", or "neither".

\(\mathbf a = \langle -4, 3, -4 \rangle\) and \(\mathbf b = \langle -20, 15, 125/4 \rangle\) are

______________________________

\(\mathbf a = \langle -4, 3, -4 \rangle\) and \(\mathbf b = \langle -20, 15, -20 \rangle\) are

______________________________

\(\mathbf a = \langle -4, 3, -4 \rangle\) and \(\mathbf b = \langle -12, 9, -13 \rangle\) are

______________________________

Show answer

\(orthogonal\), \(parallel\), \(neither\)

3

Perform the following operations on the vectors \(\vec{u} = \left\langle -4,3,-4\right\rangle\), \(\vec{v} = \left\langle 4,4,-1\right\rangle\), and \(\vec{w} = \left\langle 0,4,-3\right\rangle\).

\(\vec{u} \cdot \vec{w} =\) ______________________________

\((\vec{u} \cdot \vec{v}) \vec{u} =\) ______________________________

\(((\vec{w} \cdot \vec{w}) \vec{u} ) \cdot \vec{u} =\) ______________________________

\(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} =\) ______________________________

Show answer

\(24\), \(\left\langle0,0,0\right\rangle\), \(1025\), \(19\)

4

Find \({ \mathbf a \cdot \mathbf b }\) if \(\text{|} {\mathbf a} \text{|}\) = 2, \(\text{|} {\mathbf b} \text{|}\) = 8, and the angle between \({\mathbf a}\) and \({\mathbf b}\) is \(- \frac{\pi}{7}\) radians.

\({ \mathbf a \cdot \mathbf b }\) = ________________________________________

Show answer

\(14.4155018864387\)

5

What is the angle in radians between the vectors

\({\mathbf a}\) = (-8, 6, -7) and

\({\mathbf b}\) = (8, 8, -2)?

Angle: ____________________________________________________________ (radians)

Show answer

\(1.58505780747081\)

6

Find \({ \mathbf a \cdot \mathbf b }\) if \(\text{|} {\mathbf a} \text{|}\) = 2, \(\text{|} {\mathbf b} \text{|}\) = 8, and the angle between \({\mathbf a}\) and \({\mathbf b}\) is \(- \frac{\pi}{7}\) radians.

\({ \mathbf a \cdot \mathbf b }\) = ________________________________________

Show answer

\(14.4155018864387\)

7

A constant force \({\mathbf F} = -8 {\mathbf i} + 6 {\mathbf j} - 7 {\mathbf k}\) moves an object along a straight line from point \((7, 7, -2)\) to point \((10, 13, -11)\).

Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons.

Work: ________________________________________ Nm

Show answer

\(75\)

8

A woman exerts a horizontal force of 2 pounds on a box as she pushes it up a ramp that is 8 feet long and inclined at an angle of 30 degrees above the horizontal.

Find the work done on the box.

Work: ________________________________________ ft-lb

Show answer

\(13.856406460551\)

9

If Yoda says to Luke Skywalker, “The Force be with you,” then the dot product of the Force and Luke should be:

1. positive

2. any real number

3. zero

4. negative

Show answer

positive

10

Find the angle between the diagonal of a cube of side length 4 and the diagonal of one of its faces. The angle should be measured in radians.

______________________________

Show answer

\(0.615479708670387\)

11

Let \(\vv = \langle -2, 5 \rangle\) in \(\R^2\), and let \(\vy = \langle 0, 3, -2 \rangle\) in \(\R^3\).

1. Is \(\langle 2, -1 \rangle\) perpendicular to \(\vv\)? Why or why not?

2. Find a unit vector \(\vu\) in \(\R^2\) such that \(\vu\) is perpendicular to \(\vv\). How many such vectors are there? Justify your answers.

3. Is \(\langle 2, -1, -2 \rangle\) perpendicular to \(\vy\)? Why or why not?

4. Find a unit vector \(\vw\) in \(\R^3\) such that \(\vw\) is perpendicular to \(\vy\). How many such vectors are there?Justify your answers.

5. Let \(\vz = \langle 2, 1, 0 \rangle\). Find a unit vector \(\vr\) in \(\R^3\) such that \(\vr\) is perpendicular to both \(\vy\) and \(\vz\). How many such vectors are there? Explain your process.

12

Consider the triangle in \(\R^3\) given by \(P=(3, 2, -1)\), \(Q=(1, -2, 4)\), and \(R=(4, 4, 0)\).

1. Find the measure of each of the three angles in the triangle, accurate to \(0.01\) degrees.

2. Choose two sides of the triangle, and call the vectors that form the sides (emanating from a common point) \(\va\) and \(\vb\).

   1. Compute \(\proj_{\vb} \va\), and \(\proj_{\perp \vb} \va\).

   2. Explain why \(\proj_{\perp \vb} \va\) can be considered a height of triangle \(PQR\).

   3. Find the area of the given triangle.

13

Let \(\vu\) and \(\vv\) be vectors in \(\R^5\) with \(\vu \cdot \vv = -1\), \(| \vu | = 2\), \(| \vv | = 3\). Use the properties of the dot product to find each of the following.

1. \(\vu \cdot 2\vv\)

2. \(\vv \cdot \vv\)

3. \((\vu + \vv) \cdot \vv\)

4. \((2\vu+4\vv) \cdot (\vu - 7\vv)\)

5. \(|\vu| |\vv| \cos(\theta)\), where \(\theta\) is the angle between \(\vu\) and \(\vv\)

6. \(\theta\)

14

One of the properties of the dot product is that \((\vu+\vv) \cdot \vw = (\vu \cdot \vw) + (\vv \cdot \vw)\). That is, the dot product distributes over vector addition on the right. Here we investigate whether the dot product distributes over vector addition on the left.

1. Let \(\vu = \langle 1,2,-1 \rangle\), \(\vv = \langle 4,-3,6 \rangle\), and \(\vw = \langle 4,7,2 \rangle\). Calculate \[ \vu \cdot (\vv + \vw) \ \ \text{ and } \ \ (\vu \cdot \vv) + (\vu \cdot \vw). \] What do you notice?

2. Use the properties of the dot product to show that in general \[ \vx \cdot (\vy + \vz) = (\vx \cdot \vy) + (\vx \cdot \vz) \] for any vectors \(\vx\), \(\vy\), and \(\vz\) in \(\R^n\).

15

When running a sprint, the racers may be aided or slowed by the wind. The wind assistance is a measure of the wind speed that is helping push the runners down the track. It is much easier to run a very fast race if the wind is blowing hard in the direction of the race. So that world records aren't dependent on the weather conditions, times are only recorded as record times if the wind aiding the runners is less than or equal to 2 meters per second. Wind speed for a race is recorded by a wind gauge that is set up close to the track. It is important to note, however, that weather is not always as cooperative as we might like. The wind does not always blow exactly in the direction of the track, so the gauge must account for the angle the wind makes with the track. Suppose a 4 mile per hour wind is blowing to aid runners by making a \(38^{\circ}\) angle with the race track. Determine if any times set during such a race would qualify as records.

16

Molecular geometry is the geometry determined by arrangements of atoms in molecules. Molecular geometry includes measurements like bond angle, bond length, and torsional angles. These attributes influence several properties of molecules, such as reactivity, color, and polarity.

As an example of the molecular geometry of a molecule, consider the methane \(\text{CH}_4\) molecule, as illustrated in Figure 9.3.8. According to the Valence Shell Electron Repulsion (VSEPR) model, atoms that surround single different atoms do so in a way that positions them as far apart as possible. This means that the hydrogen atoms in the methane molecule arrange themselves at the vertices of a regular tetrahedron. The bond angle for methane is the angle determined by two consecutive hydrogen atoms and the central carbon atom. To determine the bond angle for methane, we can place the center carbon atom at the point \(\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right)\) and the hydrogen atoms at the points \((0,0,0)\), \((1,1,0)\), \((1,0,1)\), and \((0,1,1)\). Find the bond angle for methane to the nearest tenth of a degree.