| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2015 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection: show lines are skew |
| Difficulty | Challenging +1.8 This AEA question involves multiple sophisticated 3D vector techniques: proving perpendicularity and skew lines, finding perpendicular distances between skew lines, and geometric reasoning about equal distances. While each individual part uses standard methods, the extended multi-part nature requiring sustained geometric visualization and the final angle calculation from constructed points makes this significantly harder than typical A-level questions but not at the extreme difficulty level of the most challenging AEA problems. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c 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 |
The lines $L_1$ and $L_2$ have vector equations
$$L_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 10 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -5 \\ 4 \end{pmatrix}$$
$$L_2 : \mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$$
(a) Show that $L_1$ and $L_2$ are perpendicular.
[2]
(b) Show that $L_1$ and $L_2$ are skew lines.
[3]
The point $A$ with position vector $-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ lies on $L_2$ and the point $X$ lies on $L_1$ such that $\overrightarrow{AX}$ is perpendicular to $L_1$
(c) Find the position vector of $X$.
[5]
(d) Find $|\overrightarrow{AX}|$
[2]
The point $B$ (distinct from $A$) also lies on $L_2$ and $|\overrightarrow{BX}| = |\overrightarrow{AX}|$
(e) Find the position vector of $B$.
[5]
(f) Find the cosine of angle $AXB$.
[2]
\hfill \mbox{\textit{Edexcel AEA 2015 Q6 [19]}}