A-LevelFurther MathematicsVectors9231/12 · Oct/Nov 2025 · Q614 Marks
The plane Π has equation x+3y+2z = 1. Relative to O, the points A, B, C have position vectors -j+2k, 2i-k, 2i-j-k, respectively.
- (a)Find the perpendicular distance from the origin O to the plane Π.[2]
- (b)Find the acute angle between the planes OAB and Π.[4]
- (c)Find an equation for the common perpendicular to the lines OC and AB.[8]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.
- [M1 A1] Allow 0.267 or better
- [M1 A1] Finds normal to OAB. Allow 1 error for the M mark to be awarded.
- [M1] Takes dot product of normal vectors.
- [A1] Accept 0.131 radians. Mark final answer of 7.5°or better.
- [M1 A1] Finds PQ or direction of common perpendicular.
- [M1] Uses that dot product of PQ with line direction is zero, or, alternatively, PQ is a multiple of the common perpendicular [1,1,1]^T.
- [A1] Deduces one equation. (6μ - 6λ = 1)
- [A1] Deduces second equation. (14μ - 6λ = 7)
- [M1 A1] Solves for μ and substitutes into OQ and PQ or (λ=7/12, μ=3/4)
- [B1FT] OE. FT using their common perpendicular. Must have r=
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