| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Foot of perpendicular from origin to line |
| Difficulty | Moderate -0.3 Part (a) is straightforward vector magnitude calculation using scalar multiplication. Part (b) requires setting up a dot product equation using perpendicularity, which is a standard A-level technique, though the algebra involves expanding and solving a quadratic. This is a routine multi-step vectors question with no novel insight required, making it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10f Distance between points: using position vectors |
The points $O$ and $A$ have position vectors $\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 6 \\ 0 \\ 8 \end{pmatrix}$ respectively. The point $P$ is such that $\overrightarrow{OP} = k\overrightarrow{OA}$, where $k$ is a non-zero constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $k$, the length of $OP$. [1]
Point $B$ has position vector $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and angle $OPB$ is a right angle.
\item Determine the value of $k$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2023 Q2 [5]}}