What angle is between a = (4,4,4)T and b = (4,0,4)T?

A good answer might be:

Visually, it looks somewhat less than midway between perpendicular and horizontal. 35° would be a good guess.

Formula for the angle between two Vectors

One could do better than guess by noticing that in going from the tail to the head of a the vertical distance increases by 4 while the horizontal distance increases by 4 2. Hence the tangent of the angle is 4/(4 2)   =  1.0/2   =  0.7071, so the angle with the horizontal is arctan( 0.7071 )  =  35.26°. Since b is in the horizontal plane, the angle between the two vectors must be 35°.

The formula for the angle θ between two unit vectors is:

au · bu  =  cos θ

To use this formula with non-unit vectors: (1) normalize each vector, (2) compute the dot product, (3) take the arc cos to get the angle.


(Calculator Problem: ) Apply this formula to a and b of the figure.