| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Moderate -0.3 This is a straightforward multi-part vectors question testing standard techniques: scalar multiplication of vectors, finding position vectors, distance formula, angle between vectors using dot product, and perpendicularity condition. All parts follow routine procedures with no novel insight required, making it slightly easier than average for A-level. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
6 The points $A$ and $B$ have coordinates $( 2,4,1 )$ and $( 3,2 , - 1 )$ respectively. The point $C$ is such that $\overrightarrow { O C } = 2 \overrightarrow { O B }$, where $O$ is the origin.
\begin{enumerate}[label=(\alph*)]
\item Find the vectors:
\begin{enumerate}[label=(\roman*)]
\item $\overrightarrow { O C }$;
\item $\overrightarrow { A B }$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that the distance between the points $A$ and $C$ is 5 .
\item Find the size of angle $B A C$, giving your answer to the nearest degree.
\end{enumerate}\item The point $P ( \alpha , \beta , \gamma )$ is such that $B P$ is perpendicular to $A C$.
Show that $4 \alpha - 3 \gamma = 15$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2006 Q6 [12]}}