| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line of intersection of planes |
| Difficulty | Standard +0.8 This is a multi-part Further Maths vectors question requiring finding normal vectors via cross product, verifying plane equations, determining constants, and finding line of intersection in a specific non-standard form. While the individual techniques are standard FP3 material, part (d) requires converting from parametric to cross-product form which is less routine and demands careful manipulation across 5 marks. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms |
The line $l_1$ has equation
$$\mathbf{r} = \mathbf{i} + 6\mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + 3\mathbf{k})$$
and the line $l_2$ has equation
$$\mathbf{r} = 3\mathbf{i} + p\mathbf{j} + \mu(\mathbf{i} - 2\mathbf{j} + \mathbf{k}), \text{ where } p \text{ is a constant.}$$
The plane $\Pi_1$ contains $l_1$ and $l_2$.
\begin{enumerate}[label=(\alph*)]
\item Find a vector which is normal to $\Pi_1$. [2]
\item Show that an equation for $\Pi_1$ is $6x + y - 4z = 16$. [2]
\item Find the value of $p$. [1]
\end{enumerate}
The plane $\Pi_2$ has equation $\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} + \mathbf{k}) = 2$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find an equation for the line of intersection of $\Pi_1$ and $\Pi_2$, giving your answer in the form
$$(\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}.$$ [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q36 [10]}}