| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Common perpendicular to two skew lines |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring multiple sophisticated techniques: finding the common perpendicular to skew lines using the scalar triple product formula, parametric equations, simultaneous solving with perpendicularity conditions, and 3D plane geometry. While the individual steps are methodical, the overall problem requires strong spatial reasoning and coordination of several vector concepts beyond standard A-level, justifying a difficulty well above average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector4.04h Shortest distances: between parallel lines and between skew lines4.04j Shortest distance: between a point and a plane |
The line $l _ { 1 }$ passes through the point $A$ whose position vector is $3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ and is parallel to the vector $\mathbf { i } + \mathbf { j }$. The line $l _ { 2 }$ passes through the point $B$ whose position vector is $- \mathbf { i } - \mathbf { k }$ and is parallel to the vector $\mathbf { j } + 2 \mathbf { k }$. The point $P$ is on $l _ { 1 }$ and the point $Q$ is on $l _ { 2 }$ and $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$.\\
(i) Find the length of $P Q$.\\
(ii) Find the position vector of $Q$.\\
(iii) Show that the perpendicular distance from $Q$ to the plane containing $A B$ and the line $l _ { 1 }$ is $\sqrt { } 3$.
\hfill \mbox{\textit{CAIE FP1 2010 Q12 EITHER}}