Consider the vectors a \( = \sin (2\alpha )\)i \( - \cos (2\alpha )\)j + k and b \( = \cos \alpha \)i \( - \sin \alpha \)j − k, where \(0 < \alpha < 2\pi \).

Let \(\theta \) be the angle between the vectors a and b.

(a)     Express \(\cos \theta \) in terms of \(\alpha \).

(b)     Find the acute angle \(\alpha \) for which the two vectors are perpendicular.

(c)     For \(\alpha = \frac{{7\pi }}{6}\), determine the vector product of a and b and comment on the geometrical significance of this result.

(a)     \(\cos \theta = \frac{{\boldsymbol{ab}}}{{\left| \boldsymbol{a} \right|\left| \boldsymbol{b} \right|}} = \frac{{\sin 2\alpha \cos \alpha + \sin \alpha \cos 2\alpha - 1}}{{\sqrt 2 \times \sqrt 2 }}{\text{ }}\left( { = \frac{{\sin 3\alpha - 1}}{2}} \right)\)     M1A1

(b)     \({\boldsymbol{a}} \bot {\boldsymbol{b}} \Rightarrow \cos \theta = 0\)     M1

\(\sin 2\alpha \cos \alpha + \sin \alpha \cos 2\alpha - 1 = 0\)

\(\alpha = 0.524{\text{ }}\left( { = \frac{\pi }{6}} \right)\)     A1

(c)

METHOD 1

\(\left| {\begin{array}{*{20}{c}}
  {\boldsymbol{i}}&{\boldsymbol{j}}&{\boldsymbol{k}} \\
  {\sin 2\alpha }&{ - \cos 2\alpha }&1 \\
  {\cos \alpha }&{ - \sin \alpha }&{ - 1}
\end{array}} \right|\)     (M1)

assuming \(\alpha = \frac{{7\pi }}{6}\)

Note: Allow substitution at any stage.

\(\left| {\begin{array}{*{20}{c}}
  \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\
  {\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&1 \\
  { - \frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}&{ - 1}
\end{array}} \right|\)     A1

\(= \boldsymbol{i} \left( {\frac{1}{2} - \frac{1}{2}} \right) - \boldsymbol{j} \left( { - \frac{{\sqrt 3 }}{2} + \frac{{\sqrt 3 }}{2}} \right) + \boldsymbol{k}\left( {\frac{{\sqrt 3 }}{2} \times \frac{1}{2} - \frac{1}{2} \times \frac{{\sqrt 3 }}{2}} \right)\)

= 0     A1

a and b are parallel     R1

Note: Accept decimal equivalents.

METHOD 2

from (a) \(\cos \theta = - 1{\text{ (and }}\sin \theta = 0)\)     M1A1

\(\boldsymbol{a} \times \boldsymbol{b}\) = 0     A1

a and b are parallel     R1

[8 marks]

This question was attempted by most candidates who in general were able to find the dot product of the vectors in part (a). However the simplification of the expression caused difficulties which affected the performance in part (b). Many candidates had difficulties in interpreting the meaning of a \( \times \) b = 0 in part (c).