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Angle Between Two Vectors - Direction Cosines

The Law of Cosines gives us a formula for the angle between two vectors in space. In fact, we shall use the Law of Cosines to define the angle between two vectors.


Let A and B be two nonzero vectors in space. The angle between A and B is the angle θ between 0 and π such that


One can prove from the Triangle Inequality that the above quantity is always between -1 and 1, and therefore is the cosine of some angle θ (Problem 42 at the end of this section).

Example 2: Finding an Angle between Two Vectors

The direction angles of a nonzero vector A in space are the three angles α, β, γ between A and i, j, k respectively. The cosines of the direction angles are called the direction cosines of A. Let us compute the direction cosines in terms of the components of A.


The computations for β and γ are similar. Thus


Last Update: 2006-11-21