| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2004 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Common perpendicular to two skew lines |
| Difficulty | Challenging +1.8 This is a substantial Further Maths question requiring multiple vector techniques: finding the common perpendicular to skew lines (using cross product and scalar triple product), determining plane normals, calculating distances, and finding angles between planes. While the individual techniques are standard for FM students, the multi-part structure, length of working, and need to coordinate several concepts across four parts makes this significantly harder than typical A-level questions but not exceptionally difficult for Further Maths. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane4.04g Vector product: a x b perpendicular vector |
11 The line $l _ { 1 }$ passes through the point $A$, whose position vector is $3 \mathbf { i } - 5 \mathbf { j } - 4 \mathbf { k }$, and is parallel to the vector $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$. The line $l _ { 2 }$ passes through the point $B$, whose position vector is $2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }$, and is parallel to the vector $\mathbf { i } - \mathbf { j } - 4 \mathbf { k }$. The point $P$ on $l _ { 1 }$ and the point $Q$ on $l _ { 2 }$ are such that $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$. The plane $\Pi _ { 1 }$ contains $P Q$ and $l _ { 1 }$, and the plane $\Pi _ { 2 }$ contains $P Q$ and $l _ { 2 }$.\\
(i) Find the length of $P Q$.\\
(ii) Find a vector perpendicular to $\Pi _ { 1 }$.\\
(iii) Find the perpendicular distance from $B$ to $\Pi _ { 1 }$.\\
(iv) Find the angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
\hfill \mbox{\textit{CAIE FP1 2004 Q11 [12]}}