The vectors P and Q are as follows. Vector P is on the x-z plane (note that the y-value for vector P is 0) , while Q is 'behind' the y-z plane.

example

Using the formula

angle between vectors

we have:

P • Q

= (4 i + 0 j + 7 j) • (-2 i + j + 3 k )

= (4 × -2) + (0 × 1) + (7 × 3)

= 13

And now for the denominator:

|P| |Q|  

= √(42 + 02 + (7)2) × √((-2)2 + 12 + 32)

= 30.166

So

θ = arccos(13 ÷ 30.166)

Therefore the angle between the vectors P and Q is

θ = 64.47°