| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Plane containing line and point/vector |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question testing routine techniques: finding a plane equation from a point and normal vector, calculating perpendicular distance using the standard formula, and expressing a plane in parametric form. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average even for FM content. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04i Shortest distance: between a point and a line |
The plane $\Pi_1$ passes through the $P$, with position vector $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$, and is perpendicular to the line $L$ with equation
$$\mathbf{r} = 3\mathbf{i} - 2\mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}).$$
\begin{enumerate}[label=(\alph*)]
\item Show that the Cartesian equation of $\Pi_1$ is $x - 5y - 3z = -6$. [4]
\end{enumerate}
The plane $\Pi_2$ contains the line $L$ and passes through the point $Q$, with position vector $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the perpendicular distance of $Q$ from $\Pi_1$. [4]
\item Find the equation of $\Pi_2$ in the form $\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q20 [12]}}