Find

(i)     \(u \bullet v\);

(ii)     \(\left| {{u}} \right|\);

(iii)     \(\left| {{v}} \right|\).

Find the angle between \({{u}}\) and \({{v}}\).

(i)     correct substitution     (A1)

eg\(\;\;\;6 \times 2 + 3 \times 2 + 6 \times 1\)

\(u \bullet v = 24\)     A1     N2

(ii)     correct substitution into magnitude formula for \({{u}}\) or \({{v}}\)     (A1)

eg\(\;\;\;\sqrt {{6^2} + {3^2} + {6^2}} ,{\text{ }}\sqrt {{2^2} + {2^2} + {1^2}} \), correct value for \(\left| {{v}} \right|\)

\(\left| {{u}} \right| = 9\)     A1     N2

(iii)     \(\left| {{v}} \right| = 3\)     A1     N1

[5 marks]

correct substitution into angle formula     (A1)

eg\(\;\;\;\frac{{24}}{{9 \times 3}},{\text{ }}0.\bar 8\)

\(0.475882,{\text{ }}27.26604^\circ \)     A1     N2

\(0.476,{\text{ }}27.3^\circ \)

[2 marks]

Total [7 marks]

Many candidates performed well in this question. Some candidates were unfamiliar with the basis vector notation and wrongly substituted the i-j-k into the formulas. Others occasionally assumed that the magnitude could be negative.

Many candidates performed well in this question. Some candidates were unfamiliar with the basis vector notation and wrongly substituted the i-j-k into the formulas. Others occasionally assumed that the magnitude could be negative.