What angle is between a = (4,4,4)T and b = (4,0,4)T?
Visually, it looks somewhat less than midway between perpendicular and horizontal. 35° would be a good guess.
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 that value.
The formula for the angle θ between two unit vectors is:
au · bu = cosθ
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.