The scalar product can be used to find the component of one vector in the direction of another. Let x and y be nonzero vectors in either ${ℝ}^2$ or ${ℝ}^3$. We would like to write x as a sum of the form $\mathbf{p}+\mathbf{z}$, where p is in the direction of y and z is orthogonal to p (see Figure 5.1.2). To do this, let $\text{u}=(1/\Vert \mathbf{y}\Vert )\mathbf{y}$. Thus, u is a unit vector (length 1) in the

Figure 5.1.2.

direction of y. We wish to find $\alpha$ such that $\mathbf{p}=\alpha \mathbf{u}$ is orthogonal to $\mathbf{z}=\mathbf{x}-\alpha \mathbf{u}$. For p and z to be orthogonal, the scalar $\alpha$ must satisfy

The scalar α is called the scalar projection of x onto y, and the vector p is called the vector projection of x onto y.

Example 5

The point Q in Figure 5.1.3 is the point on the line $y=\frac{1}{3}x$ that is closest to the point (1, 4). Determine the coordinates of Q.

Figure 5.1.3.

Figure 5.1.3. Full Alternative Text

SOLUTION

The vector $\mathbf{\text{w}}={(3,1)}^T$ is a vector in the direction of the line $y=\frac{1}{3}x$. Let $\mathbf{v}={\left(1,4\right)}^T$. If Q is the desired point, then $Q^T$ is the vector projection of v onto w.

$$Q^T=\left({\displaystyle \frac{{\mathbf{\text{v}}}^T\mathbf{\text{w}}}{{\mathbf{\text{w}}}^T\mathbf{\text{w}}}}\right)\mathbf{\text{w}}={\displaystyle \frac{7}{10}}\left[\begin{array}{c}3\\ 1\end{array}\right]=\left[\begin{array}{c}2.1\\ 0.7\end{array}\right]$$

Thus, $Q=\left(2.1,0.7\right)$ is the closest point.

The definitions that have been given for ${ℝ}^2$ and ${ℝ}^3$ can all be generalized to ${ℝ}^n$. Indeed, if $\mathbf{x}\in {ℝ}^n$, then the Euclidean length of x is defined by

If x and y are two vectors in ${ℝ}^n$, then the distance between the vectors is $\Vert \mathbf{y}-\mathbf{x}\Vert$.

The Cauchy–Schwarz inequality holds in ${ℝ}^n$. (We will prove this in Section 5.4.) Consequently,

for any nonzero vectors x and y in ${ℝ}^n$. In view of (3), the definition of the angle between two vectors that was used for ${ℝ}^2$ can be generalized to ${ℝ}^n$. Thus, the angle $\theta$ between two nonzero vectors x and y in ${ℝ}^n$ is given by

In talking about angles between vectors, it is usually more convenient to scale the vectors so as to make them unit vectors. If we set

then the angle $\theta$ between u and v is clearly the same as the angle between x and y, and its cosine can be computed simply by taking the scalar product of the two unit vectors:

The vectors x and y are said to be orthogonal if ${\mathbf{x}}^T\mathbf{y}=0$. Often the symbol ⊥ is used to indicate orthogonality. Thus, if x and y are orthogonal, we will write $\mathbf{x}\perp \mathbf{y}$. Vector and scalar projections are defined in ${ℝ}^n$ in the same way that they were defined for ${ℝ}^2$.

If x and y are vectors in ${ℝ}^n$, then

In the case that x and y are orthogonal, equation (4) becomes the Pythagorean law

The Pythagorean law is a generalization of the Pythagorean theorem. When x and y are nonzero orthogonal vectors in ${ℝ}^2$, we can use these vectors and their sum $\mathbf{x}+\mathbf{y}$ to form a right triangle as in Figure 5.1.4. The Pythagorean law relates the lengths of the sides of the triangle. Indeed, if we set

Figure 5.1.4.

then

In many applications, the cosine of the angle between two nonzero vectors is used as a measure of how closely the directions of the vectors match up. If cos $\theta$ is near 1, then the angle between the vectors is small and hence the vectors are in nearly the same direction. A cosine value near zero would indicate that the angle between the vectors is nearly a right angle.

There are many other important applications involving angles between vectors. In particular, statisticians use the cosine of the angle between two vectors as a measure of how closely the two vectors are correlated.