Law of Cosines

Notice that when θ = π/2, cos θ = 0 and the Law of Cosines reduces to the familiar Theorem of Pythagoras, c² = a² + b².

Given vectors A and B with angle θ between them, we form a triangle with sides |A|, |B|, and |B - A|. Then by the Law of Cosines,

|B - A|² = [A|² + |B|² - 2|A||B| cos θ. Solving for cos θ, Since the arccosine is always between 0 and π,

Example 6

The direction of a vector can be described in one of three closely related ways: by its direction angles, its direction cosines, or its unit vector.

Let A be a nonzero vector. The angles α between A and i, and β between A and j, are called the direction angles of A. The cosines of these angles, cos α and cos β, are called the direction cosines of A.

The vector U = A/|A| is called the unit vector of A. U has length one, |U| = |A|/|A| = 1.

Figure 10.1.22

We can see from Figure 10.1.22 that the components of U are the direction cosines of A,

U = cos αi + cos βj.

A vector A is determined by its length and its direction cosines,

A = |A| U = |A| cos αi + |A| cos βj.

The sum of the squares of the direction cosines is always one. for

|U| = cos² α + cos² β = 1.

Example 7

Example 8