| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Triangle and parallelogram problems |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question requiring standard techniques: finding a vector between two points, calculating an angle using the scalar product formula, and using perpendicularity conditions to find coordinates. While it has multiple steps and requires careful coordinate work, all methods are routine C4 content with no novel problem-solving insight needed. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
8 The points $A$ and $B$ have coordinates $( 4 , - 2,3 )$ and $( 2,0 , - 1 )$ respectively.
The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 2 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 5 \\ - 2 \end{array} \right]$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the vector $\overrightarrow { A B }$.
\item Find the acute angle between $A B$ and the line $l$, giving your answer to the nearest degree.
\end{enumerate}\item The point $C$ lies on the line $l$ such that the angle $A B C$ is a right angle. Given that $A B C D$ is a rectangle, find the coordinates of the point $D$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2012 Q8 [12]}}