The line l₁ is parallel to the vector i – 2j – 3k and passes through the point A, whose position vector is 3i + 3j - 4k. The line l₂ is parallel to the vector –2i + j + 3k and passes through the point B, whose position vector is –3i – j + 2k. The point P on l₁ and the point Q on l₂ are such that PQ is perpendicular to both l₁ and l₂. Find (i) the length PQ, (ii) the cartesian equation of the plane Π containing PQ and l₂, (iii) the perpendicular distance of A from Π.

Q: What topic does this question test?

This question tests Vectors in A-Level Further Mathematics (syllabus code 9231). It is worth 12 marks.

Q: Which exam paper is this question from?

This question appeared in the Cambridge A-Level Further Mathematics Oct/Nov 2014 examination, Paper 1 Variant 2 (9231_w14_qp_12.pdf).

Question

The line l₁ is parallel to the vector i – 2j – 3k and passes through the point A, whose position vector is 3i + 3j - 4k. The line l₂ is parallel to the vector –2i + j + 3k and passes through the point B, whose position vector is –3i – j + 2k. The point P on l₁ and the point Q on l₂ are such that PQ is perpendicular to both l₁ and l₂. Find (i) the length PQ, (ii) the cartesian equation of the plane Π containing PQ and l₂, (iii) the perpendicular distance of A from Π.

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