| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2018 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Foot of perpendicular from general external point to line |
| Difficulty | Standard +0.8 This is a comprehensive multi-part vector question requiring coordinate geometry, angle calculations using dot products, area via cross products, vector equations, and a challenging perpendicularity condition in part (e). While individual parts use standard techniques, the combination and the final part requiring simultaneous consideration of multiple geometric constraints elevates this above average difficulty. |
| Spec | 1.10c Magnitude and direction: of vectors1.10e Position vectors: and displacement1.10f Distance between points: using position vectors4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
Question 7:
$7 \mid 38$
$147 \mid 17.479$7. The point $A$ with coordinates ( $- 3,7,2$ ) lies on a line $l _ { 1 }$ The point $B$ also lies on the line $l _ { 1 }$
Given that $\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)$,
\begin{enumerate}[label=(\alph*)]
\item find the coordinates of point $B$.
The point $P$ has coordinates ( $9,1,8$ )
\item Find the cosine of the angle $P A B$, giving your answer as a simplified surd.
\item Find the exact area of triangle $P A B$, giving your answer in its simplest form.
The line $l _ { 2 }$ passes through the point $P$ and is parallel to the line $l _ { 1 }$
\item Find a vector equation for the line $l _ { 2 }$
The point $Q$ lies on the line $l _ { 2 }$
Given that the line segment $A P$ is perpendicular to the line segment $B Q$,
\item find the coordinates of the point $Q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2018 Q7 [15]}}