| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Foot of perpendicular from general external point to line |
| Difficulty | Standard +0.8 This is a substantial multi-part 3D vectors question requiring multiple techniques: verifying a point on a line, finding vector equations, using perpendicularity conditions with dot products, and applying geometric reasoning about triangle areas. Part (d) particularly requires insight that there are two positions and involves solving a quadratic. More demanding than typical C4 questions but uses standard methods throughout. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10e Position vectors: and displacement1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
6 The points $A , B$ and $C$ have coordinates $( 3 , - 2,4 ) , ( 1 , - 5,6 )$ and $( - 4,5 , - 1 )$ respectively.
The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7 \\ - 7 \\ 5 \end{array} \right]$.
\begin{enumerate}[label=(\alph*)]
\item Show that the point $C$ lies on the line $l$.
\item Find a vector equation of the line that passes through points $A$ and $B$.
\item The point $D$ lies on the line through $A$ and $B$ such that the angle $C D A$ is a right angle. Find the coordinates of $D$.
\item The point $E$ lies on the line through $A$ and $B$ such that the area of triangle $A C E$ is three times the area of triangle $A C D$.
Find the coordinates of the two possible positions of $E$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2013 Q6 [14]}}