| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Angle between two vectors/lines (direct) |
| Difficulty | Moderate -0.3 This is a straightforward multi-part vectors question requiring standard techniques: scalar product for angle (routine calculation), vector addition to find C, then magnitude calculations. All steps are textbook exercises with no novel insight needed, making it slightly easier than average for A-level. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement |
Relative to an origin $O$, the position vectors of points $A$ and $B$ are given by
$$\overrightarrow{OA} = 2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k} \quad \text{and} \quad \overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + 4\mathbf{k}.$$
\begin{enumerate}[label=(\roman*)]
\item Use a vector method to find angle $AOB$. [4]
\end{enumerate}
The point $C$ is such that $\overrightarrow{AB} = \overrightarrow{BC}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the unit vector in the direction of $\overrightarrow{OC}$. [4]
\item Show that triangle $OAC$ is isosceles. [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2015 Q9 [9]}}