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.

Using the formula

`theta=arccos((P*Q)/(|P||Q|))`

we have:

P `*` Q= (4

i+ 0j+ 7j)`*`(−2i +j + 3k )= (4 × −2) + (0 × 1) + (7 × 3)

= 13

And now for the denominator:

`|P||Q|= sqrt (4^2+ (0)^2+ 7^2)` `xxsqrt((-2)^2+1^2+3^2) `

`= sqrt (65)sqrt(14)`

`=30.166\ "units"`

So

θ= arccos(13 ÷ 30.166)

Therefore the angle between the vectors **P** and **Q** is

θ= 64.47°