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Tim Gan Math

2020 A-Level H2 Mathematics

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Question 1Vectors4 marks

A plane \(\pi_1\) contains two vectors \(\begin{pmatrix}1\\1\\0\end{pmatrix}\) and \(\begin{pmatrix}1\\-5\\-2\end{pmatrix}\).

  1. (i) Find a vector normal to \(\pi_1\).
  2. (ii) A plane \(\pi_2\) has equation \(4x+5y-6z=0\). Find the acute angle between \(\pi_1\) and \(\pi_2\).

(i) A normal is the cross product \(\begin{pmatrix}1\\1\\0\end{pmatrix}\times\begin{pmatrix}1\\-5\\-2\end{pmatrix}=\begin{pmatrix}-2\\2\\-6\end{pmatrix}\), or any non-zero multiple of \(\begin{pmatrix}-1\\1\\-3\end{pmatrix}\).

(ii) Using normals \(\mathbf n_1=(-1,1,-3)\) and \(\mathbf n_2=(4,5,-6)\), \(\cos\theta=\frac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}=\frac{19}{\sqrt{847}}\). Hence \(\theta=49.2^\circ\) (1 d.p.).

Final Answer:

(i) \(\begin{pmatrix}-1\\1\\-3\end{pmatrix}\); (ii) \(49.2^\circ\)