| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2019 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Perpendicularity conditions |
| Difficulty | Challenging +1.8 This is an AEA question requiring geometric insight about angle bisectors and vector manipulation across multiple connected parts. Part (a) needs recognizing that the angle bisector direction is proportional to normalized vectors (a/a + b/b), then proving perpendicularity. Part (c) requires setting up equations from two conditions on D, eliminating parameters, and proving a ratio relationship. While systematic, it demands sustained multi-step reasoning and non-routine geometric vector insight beyond standard A-level, though the individual techniques are accessible. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
\begin{enumerate}
\item Points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$, respectively, relative to an origin $O$, and are such that $O A B$ is a triangle with $O A = a$ and $O B = b$.
\end{enumerate}
The point $C$, with position vector $\mathbf { c }$, lies on the line through $O$ that bisects the angle $A O B$.\\
(a) Prove that the vector $b \mathbf { a } - a \mathbf { b }$ is perpendicular to $\mathbf { c }$.
The point $D$, with position vector $\mathbf { d }$, lies on the line $A B$ between $A$ and $B$.\\
(b) Explain why $\mathbf { d }$ can be expressed in the form $\mathbf { d } = ( 1 - \lambda ) \mathbf { a } + \lambda \mathbf { b }$ for some scalar $\lambda$ with $0 < \lambda < 1$\\
(c) Given that $D$ is also on the line $O C$, find an expression for $\lambda$ in terms of $a$ and $b$ only and hence show that
$$D A : D B = O A : O B$$
\hfill \mbox{\textit{Edexcel AEA 2019 Q5 [16]}}