| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Verify perpendicularity using scalar product |
| Difficulty | Standard +0.3 This FP3 question tests basic vector geometry concepts: recognizing a line segment, applying the vector product property that parallel vectors have zero cross product, and identifying a line through the origin. While it's Further Maths content, all three parts are straightforward applications of standard definitions with no problem-solving or novel insight required—essentially recall and direct application, making it easier than average overall. |
| Spec | 1.10d Vector operations: addition and scalar multiplication4.04g Vector product: a x b perpendicular vector |
Two fixed points, $A$ and $B$, have position vectors $\mathbf{a}$ and $\mathbf{b}$ relative to the origin $O$, and a variable point $P$ has position vector $\mathbf{r}$.
\begin{enumerate}[label=(\roman*)]
\item Give a geometrical description of the locus of $P$ when $\mathbf{r}$ satisfies the equation $\mathbf{r} = \lambda\mathbf{a}$, where $0 \leq \lambda \leq 1$. [2]
\item Given that $P$ is a point on the line $AB$, use a property of the vector product to explain why $(\mathbf{r} - \mathbf{a}) \times (\mathbf{r} - \mathbf{b}) = \mathbf{0}$. [2]
\item Give a geometrical description of the locus of $P$ when $\mathbf{r}$ satisfies the equation $\mathbf{r} \times (\mathbf{a} - \mathbf{b}) = \mathbf{0}$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2008 Q3 [7]}}