| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2018 |
| Session | December |
| Marks | 5 |
| Topic | Vectors: Cross Product & Distances |
| Type | Equation of plane through three points |
| Difficulty | Standard +0.3 This is a straightforward application of standard Further Maths techniques: computing two vectors from coordinates, finding their cross product using the determinant method, then using the normal vector to write the plane equation. While it's a Further Maths topic (making it inherently harder than Core), it's a routine textbook exercise requiring no problem-solving insight—just methodical execution of learned procedures. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04g Vector product: a x b perpendicular vector |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\vec{uun} AB = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}\), \(\vec{uun} AC = \begin{pmatrix} 3 \\ 1 \\ 9 \end{pmatrix}\) | B1 | Either correct |
| \(\vec{uun} AB \times \vec{uun} AC = \begin{pmatrix} -2 \\ -3 \\ 1 \end{pmatrix}\) | M1, A1 | Cross product BC |
| (b) \(-2x - 3y + z = d\); e.g. \(-2(0) - 3(1) - 1 \times 4 = -7\); \(\Rightarrow 2x + 3y - z = 7\) oe | M1, A1 | Use of their vector product and substitution of one point |
**(a)** $\vec{uun} AB = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$, $\vec{uun} AC = \begin{pmatrix} 3 \\ 1 \\ 9 \end{pmatrix}$ | B1 | Either correct
$\vec{uun} AB \times \vec{uun} AC = \begin{pmatrix} -2 \\ -3 \\ 1 \end{pmatrix}$ | M1, A1 | Cross product BC
**(b)** $-2x - 3y + z = d$; e.g. $-2(0) - 3(1) - 1 \times 4 = -7$; $\Rightarrow 2x + 3y - z = 7$ oe | M1, A1 | Use of their vector product and substitution of one point
---1 Points $A , B$ and $C$ have coordinates $( 0,1 , - 4 ) , ( 1,1 , - 2 )$ and $( 3,2,5 )$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the vector product $\overrightarrow { A B } \times \overrightarrow { A C }$.
\item Hence find the equation of the plane $A B C$ in the form $a x + b y + c z = d$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q1 [5]}}