| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a standard multi-part vectors question requiring routine techniques: finding a line equation from two points, solving simultaneous equations for intersection, and using perpendicularity conditions. While it has multiple parts (5+ marks total), each step follows textbook methods without requiring novel insight or particularly complex algebraic manipulation. Slightly above average difficulty due to the 3D context and multi-step nature, but well within the scope of standard C4 examination questions. |
| 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 angles4.04e Line intersections: parallel, skew, or intersecting |
5 The points $A$ and $B$ have coordinates $( 5,1 , - 2 )$ and $( 4 , - 1,3 )$ respectively.\\
The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { r } - 8 \\ 5 \\ - 6 \end{array} \right] + \mu \left[ \begin{array} { r } 5 \\ 0 \\ - 2 \end{array} \right]$.
\begin{enumerate}[label=(\alph*)]
\item Find a vector equation of the line that passes through $A$ and $B$.
\item \begin{enumerate}[label=(\roman*)]
\item Show that the line that passes through $A$ and $B$ intersects the line $l$, and find the coordinates of the point of intersection, $P$.
\item The point $C$ lies on $l$ such that triangle $P B C$ has a right angle at $B$. Find the coordinates of $C$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2011 Q5 [12]}}