angle between vectors

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:

angle between vectors

Taking the numerator first, we have:

P • Q

= (3 i − 5 j) • (4 i + 6 j)

= (3 × 4) + (-5 × 6)

= -18

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

|P| |Q|  

= √(32 + (-5)2) × √(42 + 62)

= 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).