| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Triangle and parallelogram problems |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard 3D vector techniques: finding vectors from coordinates, showing a vector is a scalar multiple, calculating angles using dot product, writing line equations, and finding intersection conditions. While it has many parts (7 marks typical), each step follows routine procedures with clear guidance ('show that' questions provide the answer). The rhombus verification requires checking equal side lengths but no novel geometric insight. Slightly above average due to length and the intersection parameter finding, but all techniques are standard C4 material. |
| 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 multiplication1.10e Position vectors: and displacement1.10f Distance between points: using position vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
6
\begin{enumerate}[label=(\alph*)]
\item The points $A , B$ and $C$ have coordinates $( 3,1 , - 6 ) , ( 5 , - 2,0 )$ and $( 8 , - 4 , - 6 )$ respectively.
\begin{enumerate}[label=(\roman*)]
\item Show that the vector $\overrightarrow { A C }$ is given by $\overrightarrow { A C } = n \left[ \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right]$, where $n$ is an integer.
\item Show that the acute angle $A C B$ is given by $\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)$.
\end{enumerate}\item Find a vector equation of the line $A C$.
\item The point $D$ has coordinates $( 6 , - 1 , p )$. It is given that the lines $A C$ and $B D$ intersect.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $p$.
\item Show that $A B C D$ is a rhombus, and state the length of each of its sides.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2013 Q6 [15]}}