5.3 Linear algebra: direction cosines

Unit Vectors. Corresponding to every vector $x$ is a unit vector $u_x$ pointing in the same direction as $x$ . The term unit vector means that the norm of the vector is 1:

The question is, given $x$ , how can we find $u_x$ ? The first part of the answer is that $u_x$ will have to be a positive scalar multiple of $x$ in order to point in the same direction as $x$ , as shown in [link] . Thus

But what is $\alpha$ ? We need to choose $\alpha$ so that the norm of $u_x$ will be 1:

We have dropped the absolute value bars on $\alpha$ because $\left|\right|x\left|\right|$ is positive. The $\alpha$ that does the job is 1 over the norm of $x$ . Now we can write formulas for $u_x$ in terms of $x$ and $x$ in terms of $u_x$ :

So the vector $x$ is its direction vector $u_x$ , scaled by its Euclidean norm.

Unit Coordinate Vectors. There is a special set of unit vectors called the unit coordinate vectors. The unit coordinate vector $e_i$ is a unit vector in ${\nabla }^n$ that points in the positive direction of the $i^{th}$ coordinate axis. [link] shows the three unit coordinate vectors in ${\nabla }^3$ .

For three-dimensional space, $R^3$ , the unit coordinate vectors are

More generally, in n-dimensional space, there are $n$ unit coordinate vectors given by

You should satisfy yourself that this definition results in vectors with a norm of l.

Direction Cosines. We often say that vectors “have magnitude and direction.” This is more or less obvious from "Linear Algebra: Vectors" , where the three-dimensional vector $x$ has length $\sqrt{x_1^2+x_2^2+x_3^2}$ and points in a direction defined by the components $x_1,x_2$ , and $x_3$ . It is perfectly obvious from [link] where $x$ is written as $x=\left|\right|x\left|\right|u_x$ . But perhaps there is another representation for a vector that places the notion of magnitude and direction in even clearer evidence.

[link] shows an arbitrary vector $x\in R^3$ and the three-dimensional unit coordinate vectors $e_1,e_2,e_3$ . The inner product between the vector $x$ and the unit vector $e_k$ just reads out the $k^{th}$ component of $x$ :

Since this is true even in $R^n$ , any vector $x\in R^n$ has the following representation in terms of unit vectors:

Let us now generalize our notion of an angle $\theta$ between two vectors to $R^n$ as follows:

The celebrated Cauchy-Schwarz inequality establishes that $-1-\le$ cos $\theta \le 1$ . With this definition of angle, we may define the angle ${\theta }_k$ that a vector makes with theunit vector $e_k$ to be

But the norm of $e_k$ is 1, so

When this result is substituted into the representation of x in [link] , we obtain the formula

This formula really shows that the vector $x$ has “magnitude” $\left|\right|x\left|\right|$ and direction $({\theta }_1,{\theta }_2,...,{\theta }_n)$ and that the magnitude and direction are sufficient to determine $x$ . We call ( $cos{\theta }_1$ , cos ${\theta }_2,...$ , cos ${\theta }_n$ ) the direction cosines for the vector $x$ . In the three-dimensional case, they are illustrated in [link] .

If we compare [link] and [link] we see that the direction vector $u_x$ is composed of direction cosines:

With this definition we can write [link] compactly as

Here $x$ is written as the product of its magnitude $\left|\right|x\left|\right|$ and its direction vector $u_x$ . Now we can give an easy procedure to find a vector's direction angles:

(i) find $\left|\right|x\left|\right|$ ;

(ii) calculate $u_x=\frac{x}{\left|\right|x\left|\right|}$ ; and

(iii) take the arc cosines of the elements of $u_x$ .

Step (iii) is often unnecessary; we are usually more interested in the direction vector (unit vector) u _x . Direction vectors are used in materials science in order to study the orientation of crystal lattices and in electromagnetic fieldtheory to characterize the direction of propagation for radar and microwaves. You will find them of inestimable value in your courses on electromagneticfields and antenna design.