Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences

The goal of this section is to answer the following question. Given two vectors, what is the angle between them?

Since vectors have no position, we are free to place vectors wherever we like. If the two vectors are placed tail-to-tail, there is now a reasonable interpretation of the question: we seek the measure of the smallest angle between the two vectors, in the plane in which they lie. Figure 6.5 illustrates the situation.

Since the angle \(\theta\) lies in a triangle, we can compute it using a bit of trigonometry, namely, the law of cosines. Remember that the law of cosines states \(c^2 = a^2 + b^2 - 2ab \cos C\text{.}\)

The lengths of the sides of the triangle in Figure 6.5 are \(|\vect{v}|\text{,}\) \(|\vect{w}|\text{,}\) and \(|\vect{v}-\vect{w}|\text{.}\) Let \(\vect{v}=\langle v_1,v_2,v_3\rangle\) and \(\vect{w}=\langle w_1,w_2,w_3\rangle\text{;}\) then

A bit of simple arithmetic with the coordinates of \(\vect{v}\) and \(\vect{w}\) allows us to compute the cosine of the angle between them. If necessary we can use the arccosine to get \(\theta\text{,}\) but in many problems \(\cos\theta\) turns out to be all we really need.

The numerator of the fraction that gives us \(\cos\theta\) turns up a lot, so we give it a name and more compact notation: we call it the dot product, and write it as

This is the same symbol we use for ordinary multiplication, but there should never be any confusion; you can tell from context whether we are “multiplying” vectors or numbers. (We might also use the dot for scalar multiplication: \(a\cdot\vect{v}=a\vect{v}\text{;}\) again, it is clear what is meant from context.)

The following are some special cases worth looking at.

Generalizing the examples, note the following useful facts:

• If \(\vect{v}\) is parallel or anti-parallel to \(\vect{w}\) then \(\vect{v}\cdot\vect{w}/(|\vect{v}||\vect{w}|)=\pm1\text{,}\) and conversely, if \(\vect{v}\cdot\vect{w}/(|\vect{v}||\vect{w}|)=1\text{,}\) \(\vect{v}\) and \(\vect{w}\) are parallel, while if \(\vect{v}\cdot\vect{w}/(|\vect{v}||\vect{w}|)=-1\text{,}\) \(\vect{v}\) and \(\vect{w}\) are anti-parallel. (Vectors are parallel if they point in the same direction, anti-parallel if they point in opposite directions.)

• If \(\vect{v}\) is perpendicular to \(\vect{w}\) then \(\vect{v}\cdot\vect{w}/(|\vect{v}||\vect{w}|)=0\text{,}\) and conversely if \(\vect{v}\cdot\vect{w}/(|\vect{v}||\vect{w}|)=0\) then \(\vect{v}\) and \(\vect{w}\) are perpendicular.

Given two vectors, it is often useful to find the projection of one vector onto the other, because this turns out to have important meaning in many circumstances. More precisely, given \(\vect{v}\) and \(\vect{w}\text{,}\) we seek a vector parallel to \(\vect{w}\) but with length determined by \(\vect{v}\) in a natural way, as shown in Figure 6.6. \(\vect{p}\) is chosen so that the triangle formed by \(\vect{v}\text{,}\) \(\vect{p}\text{,}\) and \(\vect{v}-\vect{p}\) is a right triangle.

Using a little trigonometry, we see that

this is sometimes called the scalar projection of \(\vect{v}\) onto \(\vect{w}\). To get \(\vect{p}\) itself, we multiply this length by a vector of length one parallel to \(\vect{w}\text{:}\)

Be sure that you understand why \(\vect{w}/|\vect{w}|\) is a vector of length one (also called a unit vector ) parallel to \(\vect{w}\text{.}\)

The discussion so far implicitly assumed that \(0\le\theta\le\pi/2\text{.}\) If \(\pi/2\lt \theta\le\pi\text{,}\) the picture is like Figure 6.7. In this case \(\vect{v}\cdot \vect{w}\) is negative, so the vector

is anti-parallel to \(\vect{w}\text{,}\) and its length is

In general, the scalar projection of \(\vect{v}\) onto \(\vect{w}\) may be positive or negative. If it is negative, it means that the projection vector is anti-parallel to \(\vect{w}\) and that the length of the projection vector is the absolute value of the scalar projection. Of course, you can also compute the length of the projection vector as usual, by applying the distance formula to the vector.

Note that the phrase “projection onto \(\vect{w}\)” is a bit misleading if taken literally; all that \(\vect{w}\) provides is a direction; the length of \(\vect{w}\) has no impact on the final vector. In Figure 6.8, for example, \(\vect{w}\) is shorter than the projection vector, but this is perfectly acceptable.

Physical force is a vector quantity. It is often necessary to compute the “component” of a force acting in a different direction than the force is being applied.

The dot product has some familiar-looking properties that will be useful later, so we list them here. These may be proved by writing the vectors in coordinate form and then performing the indicated calculations; subsequently it can be easier to use the properties instead of calculating with coordinates.