| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2008 |
| Session | June |
| Marks | 22 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Angle between two vectors/lines (direct) |
| Difficulty | Challenging +1.8 This AEA question requires multiple vector techniques including angle calculation, angle bisector derivation, and finding an inscribed circle (incircle) in 3D. While parts (a)-(c) are standard A-level vector operations, parts (d)-(e) require understanding that the incircle center lies on the angle bisector and using perpendicular distance formulas from point to plane, which demands geometric insight and extended multi-step reasoning beyond typical A-level fare. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04a Line equations: 2D and 3D, cartesian and vector forms4.04j Shortest distance: between a point and a plane |
Relative to a fixed origin $O$, the position vectors of the points $A$, $B$ and $C$ are
$$\overrightarrow{OA} = -3\mathbf{i} + \mathbf{j} - 9\mathbf{k}, \quad \overrightarrow{OB} = \mathbf{i} - \mathbf{k}, \quad \overrightarrow{OC} = 5\mathbf{i} + 2\mathbf{j} - 5\mathbf{k} \text{ respectively}.$$
\begin{enumerate}[label=(\alph*)]
\item Find the cosine of angle $ABC$. [4]
\end{enumerate}
The line $L$ is the angle bisector of angle $ABC$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that an equation of $L$ is $\mathbf{r} = \mathbf{i} - \mathbf{k} + t(\mathbf{i} + 2\mathbf{j} - 7\mathbf{k})$. [4]
\item Show that $|\overrightarrow{AB}| = |\overrightarrow{AC}|$. [2]
\end{enumerate}
The circle $S$ lies inside triangle $ABC$ and each side of the triangle is a tangent to $S$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the position vector of the centre of $S$. [7]
\item Find the radius of $S$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2008 Q7 [22]}}