A-LevelFurther MathematicsVectors9231/12 · Oct/Nov 2021 · Q717 Marks

The points A, B, C have position vectors 2i+2j, -j+k and 2i+j-7k respectively, relative to the origin O.

(a)Find an equation of the plane OAB, giving your answer in the form r.n = p.[3]
(b)The plane Π has equation x-3y-2z = 1. Find the perpendicular distance of Π from the origin.[1]
(c)Find the acute angle between the planes OAB and Π.[3]
(d)Find an equation for the common perpendicular to the lines OC and AB.[10]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[M1 A1] (a) Finds common perpendicular.
[A1] (a) r.(-1,1,1)=0
[B1] (b) Divides by magnitude of the normal to Π 0.267
[M1] (c) Takes dot product of normal vectors.
[A1 FT] (c) 6/√14
[A1] (c) Accept 0.388 radians. Mark final answer.
[M1 A1] (d) Finds direction of common perpendicular.
[M1 A1] (d) Finds PQ.
[M1] (d) Uses that dot product of PQ with line direction is zero, or, alternatively, PQ is a multiple of the common perpendicular (parameter k not 1).
[A1] (d) Deduces one equation.
[A1] (d) Deduces second equation.
[M1 A1] (d) Solves for λ and substitutes into OP.
[B1 FT] (d) FT using their common perpendicular.

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