| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Show lines are skew (non-intersecting) |
| Difficulty | Standard +0.3 This FP3 question tests basic vector geometry concepts: recognizing line segments, understanding when vector products are zero (parallel vectors), and interpreting vector equations geometrically. All parts require recall of standard properties rather than problem-solving or novel insight. While it's Further Maths content, these are foundational vector product concepts that are relatively straightforward once the theory is known. |
| Spec | 4.04g 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 $0 \leqslant \lambda \leqslant 1$. [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 Q3 [7]}}