Vectors – Year 11 GCSE maths test to download as PDF

\(\mathbf a=\begin{pmatrix}2\\3\end{pmatrix}\) and \(\mathbf b=\begin{pmatrix}-1\\5\end{pmatrix}\).

1. Work out \(2\mathbf a-\mathbf b\). (2 marks)
2. Work out the magnitude of \(\mathbf a+\mathbf b\), to 2 decimal places. (2 marks)

A boat sails from \(P\) to \(Q\) with \(\overrightarrow{PQ}=\begin{pmatrix}6\\-2\end{pmatrix}\) km, and then from \(Q\) to \(R\) with \(\overrightarrow{QR}=\begin{pmatrix}-10\\5\end{pmatrix}\) km.

1. Find \(\overrightarrow{PR}\). (2 marks)
2. Find the distance \(PR\) to 1 decimal place. (2 marks)

\(\mathbf p=\begin{pmatrix}3\\-2\end{pmatrix}\), \(\mathbf q=\begin{pmatrix}9\\k\end{pmatrix}\) and \(\mathbf r=\begin{pmatrix}-6\\4\end{pmatrix}\).

1. Show that \(\mathbf p\) and \(\mathbf r\) are parallel. (2 marks)
2. \(\mathbf p\) and \(\mathbf q\) are parallel. Find \(k\). (2 marks)

\(OAB\) is a triangle with \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). \(M\) is the midpoint of \(OA\). \(N\) is on \(AB\) with \(AN:NB=1:3\).

1. Find \(\overrightarrow{AB}\). (1 mark)
2. Show that \(\overrightarrow{ON}=\tfrac34\mathbf a+\tfrac14\mathbf b\). (2 marks)
3. Find \(\overrightarrow{MN}\). (1 mark)

A helicopter flies from a base \(B\) to a ship \(S\) where \(\overrightarrow{BS}=\begin{pmatrix}24\\10\end{pmatrix}\) km. It then flies on to an oil rig \(R\) where \(\overrightarrow{SR}=\begin{pmatrix}-8\\-16\end{pmatrix}\) km.

1. Find the vector \(\overrightarrow{BR}\). (1 mark)
2. Work out the total distance flown from \(B\) to \(R\) via \(S\), to 1 decimal place. (2 marks)
3. The helicopter has fuel for 60 km. Can it return directly from \(R\) to \(B\) after flying via \(S\)? Show your working. (1 mark)