A-LevelMathematicsVectors9709/33 · Oct/Nov 2022 · Q99 Marks

With respect to the origin O, the position vectors of the points A, B and C are given by OA = (0, 5, 2), OB = (1, 0, 1) and OC = (4, -3, -2). The midpoint of AC is M and the point N lies on BC, between B and C, and is such that BN = 2NC.

(a)Find the position vectors of M and N.[3]
(b)Find a vector equation for the line through M and N.[2]
(c)Find the position vector of the point Q where the line through M and N intersects the line through A and B.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

B1 State OM = (2, 1, 0)
M1 Use a correct method to find ON
A1 Obtain answer (3, -2, -1)
M1 Carry out a correct method to form a vector equation for MN
A1 OE Obtain a correct equation in any form, e.g. r = (2, 1, 0) + λ(1, -3, -1)
B1 State a correct vector equation for AB in any form, e.g. r = (0, 5, 2) + μ(1, -5, -1)
M1 Equate components of AB and MN and solve for λ or for μ
A1 Obtain λ = – 3 or μ = -1
A1 Obtain position vector (3, -5, 3), or equivalent, for Q

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