Find the cosine of the angle between the two vectors \(3{\boldsymbol{i}} + 4{\boldsymbol{j}} + 5{\boldsymbol{k}}\) and \(4{\boldsymbol{i}} - 5{\boldsymbol{j}} - 3{\boldsymbol{k}}\) .

finding scalar product and magnitudes     (A1)(A1)(A1)

scalar product \( = 12 - 20 - 15\) (\( = - 23\)) 

magnitudes \( = \sqrt {{3^2} + {4^2} + {5^2}} \) , \( = \sqrt {{4^2} + {{( - 5)}^2} + {{( - 3)}^2}} \) , \(\left( {\sqrt {50} ,\sqrt {50} } \right)\)

substitution into formula     M1

e.g. \(\cos \theta  = \frac{{12 - 20 - 15}}{{\left( {\sqrt {{3^2} + {4^2} + {5^2}} } \right) \times \left( {\sqrt {{4^2} + {{( - 5)}^2} + {{( - 3)}^2}} } \right)}}\)

\(\cos \theta  = - \frac{{23}}{{50}}\) \(( = - 0.46)\)     A2     N4

[6 marks]

Many candidates performed well in finding the magnitudes and scalar product to use the formula for angle between vectors. Some experienced trouble with the arithmetic to obtain the required result. A significant number of candidates isolated the theta answering with \({\text{arccos}}\left( {\frac{{ - 23}}{{50}}} \right)\) .