Direction Cosines

This application of the dot product needs that we be in three dimensional (3D) space not like all the other applications we have looked at to this point. 

Let us begin with a vector, a^→ in three dimensional (3D) space.  This vector will make angles along with the x-axis (α), the y-axis (β), and the z-axis (γ).  These angles are known as direction angles and the cosines of these angles are known as direction cosines.

Here is a diagram of a vector and the direction angles.

The formulas for the direction cosines are as follow:

cos α = a^→• i^→ / ||a^→|| = a₁ / ||a^→||

cos β = a • j / ||a^→|| = a₂ / ||a^→||

cos γ = a • k / ||a^→|| = a₃ / ||a^→||

where i, j, and k are the standard basis vectors.