| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2006 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Common perpendicular to two skew lines |
| Difficulty | Challenging +1.8 This is a multi-part AEA question on skew lines requiring: (a) showing non-intersection via solving a 3-equation system with no solution, (b) verifying perpendicularity via dot products (routine), and (c) finding the common perpendicular by setting up and solving simultaneous equations with the perpendicular direction vector. While it requires careful algebraic manipulation and understanding of 3D geometry, the techniques are systematic and the perpendicular vector is given in part (b), reducing the conceptual demand. Harder than standard A-level but not requiring exceptional insight. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting4.04g Vector product: a x b perpendicular vector4.04h Shortest distances: between parallel lines and between skew lines |
5.The lines $L _ { 1 }$ and $L _ { 2 }$ have vector equations\\
$L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )$,\\
$L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )$,\\
where $\lambda$ and $\mu$ are parameters.
\begin{enumerate}[label=(\alph*)]
\item Show that $L _ { 1 }$ and $L _ { 2 }$ do not intersect.
\item Show that the vector $( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )$ is perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ .
The point $A$ lies on $L _ { 1 }$ ,the point $B$ lies on $L _ { 2 }$ and $A B$ is perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ .
\item Find the position vector of the point $A$ and the position vector of the point $B$ .\\
(8)\\
\includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571}
Figure 1 shows a sketch of part of the curve $C$ with equation
$$y = \sin ( \ln x ) , \quad x \geq 1 .$$
The point $Q$ ,on $C$ ,is a maximum.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2006 Q5 [15]}}