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 35°.

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

a= cos θ_{u}· b_{u}

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.