In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 4 units and OG = 2 units. Unit vectors i, j and k are parallel to OA, OC and OG respectively. The point M is the midpoint of DF. The point N on AB is such that AN = 3NB. [Figure 9.1]

Mark scheme

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

1. [B1] Obtain OM=i+2j+2k
2. [M1] Use a correct method to find MN e.g. MO + OA + AN or MO + ON
3. [A1] Obtain MN=i+j-2k Accept any notation.
4. [M1] Use a correct method to form an equation for MN Allow without r = ...
5. [A1 FT] Obtain r = 2i + 3j + 1 (i + j – 2k) OE e.g. r = i + 2j + 2k + μ (i + j − 2k ) Must have r = ... Follow their answers to part 9(a).
6. [B1] State OP for a general point P on MN in component form, e.g. (2 + λ, 3 + λ,−2λ)
7. [M1] Equate scalar product of OP and a direction vector for MN to zero and solve for λ
8. [A1] Obtain λ = 5/6 OE e.g. μ= -1/6
9. [A1] Obtain √(53/6) correctly AG e.g. from √((2/6)²+(28/6)²+(2/6)²)