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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.
Note on point of tangent
Cross Product In this last section we will look at the cross product of two vectors. We must note that the cross product needs both of the vectors to be three dimensional (3D
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