| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Cartesian equation of a plane |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard vector and plane techniques (finding direction vectors, verifying perpendicularity, writing plane equations, finding angles and intersections). All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane |
| Answer | Marks |
|---|---|
| \(\overrightarrow{AB} = \begin{pmatrix}-2\\-1\\-1\end{pmatrix}, \overrightarrow{AC} = \begin{pmatrix}-1\\-11\\3\end{pmatrix}\) | B1 B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{n}.\overrightarrow{AB} = \begin{pmatrix}2\\-1\\-3\end{pmatrix}.\begin{pmatrix}-2\\-1\\-1\end{pmatrix} = -4+1+3 = 0\) | M1, E1 | scalar product |
| \(\mathbf{n}.\overrightarrow{AC} = \begin{pmatrix}2\\-1\\-3\end{pmatrix}.\begin{pmatrix}-1\\-11\\3\end{pmatrix} = -2+11-9 = 0\) | E1 | |
| plane is \(2x - y - 3z = d\) | M1 | |
| \(x=1, y=3, z=-2 \Rightarrow d = 2-3+6 = 5\) | ||
| plane is \(2x - y - 3z = 5\) | A1 [5] |
# Question 2:
## Part (i):
$\overrightarrow{AB} = \begin{pmatrix}-2\\-1\\-1\end{pmatrix}, \overrightarrow{AC} = \begin{pmatrix}-1\\-11\\3\end{pmatrix}$ | B1 B1 [2] |
## Part (ii):
$\mathbf{n}.\overrightarrow{AB} = \begin{pmatrix}2\\-1\\-3\end{pmatrix}.\begin{pmatrix}-2\\-1\\-1\end{pmatrix} = -4+1+3 = 0$ | M1, E1 | scalar product
$\mathbf{n}.\overrightarrow{AC} = \begin{pmatrix}2\\-1\\-3\end{pmatrix}.\begin{pmatrix}-1\\-11\\3\end{pmatrix} = -2+11-9 = 0$ | E1 |
plane is $2x - y - 3z = d$ | M1 |
$x=1, y=3, z=-2 \Rightarrow d = 2-3+6 = 5$ | |
plane is $2x - y - 3z = 5$ | A1 [5] |
---2 The points $\mathrm { A } , \mathrm { B }$ and C have coordinates $( 1,3 , - 2 ) , ( - 1,2 , - 3 )$ and $( 0 , - 8,1 )$ respectively.\\
(i) Find the vectors $\overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { AC } }$.\\
(ii) Show that the vector $2 \mathbf { i } - \mathbf { j } - 3 \mathbf { k }$ is perpendicular to the plane ABC . Hence find the equation of the plane ABC .\\
(i) Write down normal vectors to the planes $2 x - y + z = 2$ and $x - z = 1$.
Hence find the acute angle between the planes.\\
(ii) Write down a vector equation of the line through $( 2,0,1 )$ perpendicular to the plane $2 x - y + z = 2$. Find the point of intersection of this line with the plane.\\
(i) Find the cartesian equation of the plane through the point $( 2 , - 1,4 )$ with normal vector
$$\mathbf { n } = \left( \begin{array} { l }
1 \\
1 \\
2
\end{array} \right) .$$
(ii) Find the coordinates of the point of intersection of this plane and the straight line with equation
$$\mathbf { r } = \left( \begin{array} { r }
7 \\
12 \\
9
\end{array} \right) + \lambda \left( \begin{array} { l }
1 \\
3 \\
2
\end{array} \right)$$
\hfill \mbox{\textit{OCR MEI C4 Q2 [7]}}