| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | February |
| Marks | 11 |
| Topic | Vectors: Cross Product & Distances |
| Type | Area of triangle using cross product |
| Difficulty | Standard +0.3 This is a standard Further Maths 3D vectors question requiring routine techniques: finding a line equation from two points, using perpendicularity to find the closest point, and calculating triangle area using the cross product. All methods are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04g Vector product: a x b perpendicular vector |
Points $A$, $B$ and $C$ have coordinates $(4, 2, 0)$, $(1, 5, 3)$ and $(1, 4, -2)$ respectively.
The line $l$ passes through $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Find a cartesian equation for $l$. [3]
\end{enumerate}
$M$ is the point on $l$ that is closest to $C$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the coordinates of $M$. [4]
\item Find the exact area of the triangle $ABC$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q5 [11]}}