A-LevelMathematicsVectors9709/32 · Feb/Mar 2020 · Q89 Marks

In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 3 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OC and OD respectively. The point M on AB is such that MB = 2AM. The midpoint of FG is N. [Figure 8.1]

(a)Express the vectors OM and MN in terms of i, j and k.[3]
(b)Find a vector equation for the line through M and N.[2]
(c)Find the position vector of P, the foot of the perpendicular from D to the line through M and N.[4]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.

[B1] Obtain OM = 2i + j
[M1] Use a correct method to find MN
[A1] Obtain MN = − i + 2j + 2k
[M1] Use a correct method to form an equation for MN
[A1] Obtain r = 2i + j + λ(− i + 2j + 2k), or equivalent
[B1] Find DP for a point P on MN with parameter λ, e.g. (2−λ, 1+2λ, −2+2λ)
[M1] Equate scalar product of DP and a direction vector for MN to zero and solve for λ
[A1] Obtain λ = 4/9
[A1] State that the position vector of P is 14/9i + 17/9j + 8/9k

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