9 The plane \(\Pi _ { 1 }\) has parametric equation
$$\mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$
Find a cartesian equation of \(\Pi _ { 1 }\).
The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - 2 y - 3 z = 4\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
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Question 9:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & -1 \\ 1 & 2 & -2 \end{vmatrix} = \begin{pmatrix} 6 \\ 1 \\ 4 \end{pmatrix}\) M1A1
Finds normal vector to plane
Plane equation \(6x + y + 4z =\) constant; substitute \((2,-3,1) \Rightarrow 12-3+4=13\) M1
Uses known point to find constant term
\(\Rightarrow \Pi_1: 6x + y + 4z = 13\) A1
4 marks total
\(\cos\theta = \frac{(6\mathbf{i}+\mathbf{j}+4\mathbf{k})\cdot(3\mathbf{i}-2\mathbf{j}-3\mathbf{k})}{\sqrt{6^2+1^2+4^2}\sqrt{3^2+2^2+3^2}} = \frac{4}{\sqrt{53}\sqrt{22}}\) M1A1
Angle between normals equals angle between planes
\(\Rightarrow \theta = 83.3°\) or \(1.45\) rad A1
3 marks total
\(6x+y+4z=13\) and \(3x-2y-3z=4\); obtains e.g. \(y+2z=1\) and \(3x+z=6\) M1
Solve plane equations simultaneously
Or two of \((0,-11,6)\), \((11/6, 0, 1/2)\), \((2,1,0)\) A1A1
\(\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}2\\1\\0\end{pmatrix} + t\begin{pmatrix}-1\\-6\\3\end{pmatrix}\) A1
4 marks total
Alternatively: Direction of line from vector product (M1A1); finds a point on line (A1); states equation of line (A1)
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## Question 9:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & -1 \\ 1 & 2 & -2 \end{vmatrix} = \begin{pmatrix} 6 \\ 1 \\ 4 \end{pmatrix}$ | M1A1 | Finds normal vector to plane |
| Plane equation $6x + y + 4z =$ constant; substitute $(2,-3,1) \Rightarrow 12-3+4=13$ | M1 | Uses known point to find constant term |
| $\Rightarrow \Pi_1: 6x + y + 4z = 13$ | A1 | 4 marks total |
| $\cos\theta = \frac{(6\mathbf{i}+\mathbf{j}+4\mathbf{k})\cdot(3\mathbf{i}-2\mathbf{j}-3\mathbf{k})}{\sqrt{6^2+1^2+4^2}\sqrt{3^2+2^2+3^2}} = \frac{4}{\sqrt{53}\sqrt{22}}$ | M1A1 | Angle between normals equals angle between planes |
| $\Rightarrow \theta = 83.3°$ or $1.45$ rad | A1 | 3 marks total |
| $6x+y+4z=13$ and $3x-2y-3z=4$; obtains e.g. $y+2z=1$ and $3x+z=6$ | M1 | Solve plane equations simultaneously |
| Or two of $(0,-11,6)$, $(11/6, 0, 1/2)$, $(2,1,0)$ | A1A1 | |
| $\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}2\\1\\0\end{pmatrix} + t\begin{pmatrix}-1\\-6\\3\end{pmatrix}$ | A1 | 4 marks total |
**Alternatively:** Direction of line from vector product (M1A1); finds a point on line (A1); states equation of line (A1)
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9 The plane $\Pi _ { 1 }$ has parametric equation
$$\mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$
Find a cartesian equation of $\Pi _ { 1 }$.
The plane $\Pi _ { 2 }$ has cartesian equation $3 x - 2 y - 3 z = 4$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
Find a vector equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
\hfill \mbox{\textit{CAIE FP1 2012 Q9 [11]}}