A-LevelMathematicsVectors9709/33 · Oct/Nov 2019 · Q79 Marks

The plane m has equation x + 4y – 8z = 2. The plane n is parallel to m and passes through the point P with coordinates (5, 2, −2).

(i)Find the equation of n, giving your answer in the form ax + by + cz = d.[2]
(ii)Calculate the perpendicular distance between m and n.[3]
(iii)The line l lies in the plane n, passes through the point P and is perpendicular to OP, where O is the origin. Find a vector equation for l.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[M1] Substitute coordinates (5, 2, – 2) in x + 4y – 8z = d
[A1] Obtain plane equation x + 4y – 8z = 29, or equivalent
[M1] Attempt to use perpendicular formula to find perpendicular from (5, 2, – 2) to m
[A1] Obtain a correct unsimplified expression, e.g. (5+8+16-2) / √(1+16+64)
[A1] Obtain answer 3
[B1] Calling the direction vector ai + bj + ck, use a scalar product to form a relevant equation in a, b and c, e.g. a + 4b – 8c = 0 or 5a + 2b – 2c = 0
[M1] Solve two relevant equations for the ratio a : b : c
[A1] Obtain a : b : c = 4 : -19 : –9
[A1] State answer r = 5i + 2j – 2k + λ(4i – 19j – 9k)

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