| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Plane containing line and point/vector |
| Difficulty | Standard +0.8 This FP3 question requires finding planes containing skew lines, using cross products to find normal vectors, and recognizing the geometric relationship between parallel planes and skew lines. While the individual techniques are standard (cross products, plane equations), the multi-part structure requiring conceptual understanding of 3D geometry and the final insight about skew line distance makes it moderately challenging but still within typical Further Maths scope. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04i Shortest distance: between a point and a line |
Lines $l_1$ and $l_2$ have equations
$$\frac{x-3}{2} = \frac{y-4}{-1} = \frac{z+1}{1} \quad \text{and} \quad \frac{x-5}{4} = \frac{y-1}{3} = \frac{z-1}{2}$$
respectively.
\begin{enumerate}[label=(\roman*)]
\item Find the equation of the plane $\Pi_1$ which contains $l_1$ and is parallel to $l_2$, giving your answer in the form $\mathbf{r} \cdot \mathbf{n} = p$. [5]
\item Find the equation of the plane $\Pi_2$ which contains $l_2$ and is parallel to $l_1$, giving your answer in the form $\mathbf{r} \cdot \mathbf{n} = p$. [2]
\item Find the distance between the planes $\Pi_1$ and $\Pi_2$. [2]
\item State the relationship between the answer to part (iii) and the lines $l_1$ and $l_2$. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q6 [10]}}