Standard +0.8 This is a substantial three-part Further Maths question requiring: (1) finding a plane equation from three points using cross product of direction vectors, (2) calculating angle between planes using normal vectors, and (3) finding line of intersection by solving simultaneous equations and expressing parametrically. While each technique is standard for FM students, the combination of multiple steps and the need for accurate vector manipulation throughout makes this moderately challenging.
8 Find a cartesian equation of the plane \(\Pi _ { 1 }\) passing through the points with coordinates \(( 2 , - 1,3 )\), \(( 4,2 , - 5 )\) and \(( - 1,3 , - 2 )\).
The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - y + 2 z = 5\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
Find a vector equation of the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
\(\Rightarrow \theta = 70.9°\) or \(1.24\) radians
A1 [3]
Direction of line of intersection is \(\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\1&2&1\\3&-1&2\end{vmatrix} = \begin{pmatrix}5\\1\\-7\end{pmatrix}\)
M1A1
Finds point common to both planes e.g. \((-1,0,4)\) or \(\left(\frac{13}{7},\frac{4}{7},0\right)\) or \(\left(0,\frac{1}{5},\frac{13}{5}\right)\)
M1
Equation of line of intersection is e.g. \(\mathbf{r} = \begin{pmatrix}-1\\0\\4\end{pmatrix} + t\begin{pmatrix}5\\1\\-7\end{pmatrix}\)
A1\(\checkmark\) [4]
# Question 8:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\overrightarrow{AB} \times \overrightarrow{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & -8 \\ -3 & 4 & -5 \end{vmatrix} = \begin{pmatrix} 17 \\ 34 \\ 17 \end{pmatrix} \sim \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ | M1 A1 | |
| $x + 2y + z = \text{const} \Rightarrow \text{const} = 2-2+3 = 3$ (using a point) $\Rightarrow x+2y+z=3$ | M1, A1 [4] | |
| $\sqrt{9+1+4}\sqrt{1+4+1}\cos\theta = \begin{pmatrix}1\\2\\1\end{pmatrix}\cdot\begin{pmatrix}3\\-1\\2\end{pmatrix} \Rightarrow \cos\theta = \frac{3}{\sqrt{14}\sqrt{6}} = \frac{3}{\sqrt{84}}$ | M1 M1 | |
| $\Rightarrow \theta = 70.9°$ or $1.24$ radians | A1 [3] | |
| Direction of line of intersection is $\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\1&2&1\\3&-1&2\end{vmatrix} = \begin{pmatrix}5\\1\\-7\end{pmatrix}$ | M1A1 | |
| Finds point common to both planes e.g. $(-1,0,4)$ or $\left(\frac{13}{7},\frac{4}{7},0\right)$ or $\left(0,\frac{1}{5},\frac{13}{5}\right)$ | M1 | |
| Equation of line of intersection is e.g. $\mathbf{r} = \begin{pmatrix}-1\\0\\4\end{pmatrix} + t\begin{pmatrix}5\\1\\-7\end{pmatrix}$ | A1$\checkmark$ [4] | |
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8 Find a cartesian equation of the plane $\Pi _ { 1 }$ passing through the points with coordinates $( 2 , - 1,3 )$, $( 4,2 , - 5 )$ and $( - 1,3 , - 2 )$.
The plane $\Pi _ { 2 }$ has cartesian equation $3 x - y + 2 z = 5$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
Find a vector equation of the line of intersection of the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
\hfill \mbox{\textit{CAIE FP1 2016 Q8 [11]}}