| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2009 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Angle between vectors using scalar product |
| Difficulty | Challenging +1.8 This AEA question requires multiple sophisticated techniques: finding angles via scalar product, computing kite area using cross products, determining an inscribed circle radius (requiring geometric insight about kite properties and tangent lengths), and finding the fourth vertex. The fractional coordinates add computational complexity, and part (c)'s specific form requirement demands algebraic manipulation. While systematic, it requires sustained multi-step reasoning across geometry and vectors beyond standard A-level. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
7.Relative to a fixed origin $O$ the points $A , B$ and $C$ have position vectors
$$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$
\begin{enumerate}[label=(\alph*)]
\item Find the cosine of angle $A B C$ .
The quadrilateral $A B C D$ is a kite $K$ .
\item Find the area of $K$ .
A circle is drawn inside $K$ so that it touches each of the 4 sides of $K$ .
\item Find the radius of the circle,giving your answer in the form $p \sqrt { } ( q ) - q \sqrt { } ( p )$ ,where $p$ and $q$ are positive integers.
\item Find the position vector of the point $D$ .\\
(Total 18 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2009 Q7 [18]}}