We aim to find angle POQ. We observe that it is an obtuse angle (between 90° and 180°).

We use the formula we just derived:

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

Taking the numerator first, we have:

P `*` Q= (3

i− 5j)•(4i +6j)= (3 × 4) + (-5 × 6)

= -18

And now for the denominator (the magnitudes are found using Pythagoras' Theorem):

|P| |Q|^{ }= √(3

^{2}+ (-5)^{2}) × √(4^{2}+ 6^{2})= 42.048

So

θ= arccos(−18 ÷ 42.048)

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

θ= 115.3°

This looks a reasonable answer considering the diagram above (which is drawn to scale).

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