Given that \({\boldsymbol{a}} = 2\sin \theta {\boldsymbol{i}} + \left( {1 - \sin \theta } \right){\boldsymbol{j}}\) , find the value of the acute angle \(\theta \) , so that \(\boldsymbol{a}\) is perpendicular to the line \(x + y = 1\).

direction vector for line \( = \left( {\begin{array}{*{20}{c}}
  1 \\
  { - 1}
\end{array}} \right)\) or any multiple     A1

\(\left( {\begin{array}{*{20}{c}}
  {2\sin \theta } \\
  {1 - \sin \theta }
\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}
  1 \\
  { - 1}
\end{array}} \right) = 0\)     M1

\(2\sin \theta  - 1 + \sin \theta  = 0\)     A1

Note: Allow FT on candidate’s direction vector just for line above only.

\(3\sin \theta  = 1\)

\(\sin \theta  = \frac{1}{3}\)     A1

\(\theta  = 0.340\) or \(19.5\)     A1

Note: A coordinate geometry method using perpendicular gradients is acceptable.

[5 marks]

A variety of approaches were seen, either using a scalar product of vectors, or based on the rule for perpendicular gradients of lines. The main problem encountered in the first approach was in the choice of the correct vector direction for the line.