The quadrilateral ABCD is a trapezium in which AB and DC are parallel. With respect to the origin O, the position vectors of A, B and C are given by OA = −i + 2j + 3k, OB = i + 3j + k and OC = 2i + 2j – 3k.

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

1. [B1] State or imply AB = 2i + j - 2k OE
2. [M1] Carry out a correct method to find OD
3. [A1] Obtain answer -4i-j+3k OE
4. [B1FT] State r=-i+2j+3k + λ(2i + j−2k)
5. [*M1] For a general point P on AB, state CP or DP in component form, e.g. CP = (3 – 2λ, −λ,−6+2λ)
6. [DM1] Equate a relevant scalar product to zero or equate derivative of |CP| to zero or use Pythagoras in a relevant triangle and solve for λ
7. [A1] Obtain λ = 2
8. [A1] Show the perpendicular is of length 3
9. [A1] Carry out a correct method to find the area of ABCD and obtain the answer 18
10. [*M1] (Alternative method for Question 9(c)) Use a scalar product to find the projection CN (or DN) of BC (or AD) on CD
11. [A1] (Alternative method) Obtain CN = 3 (or DN = 3)
12. [DM1] (Alternative method) Use Pythagoras to obtain BN (or AN)
13. [A1] (Alternative method) Obtain answer 3
14. [A1] (Alternative method) Carry out a correct method to find the area of ABCD and obtain the answer 18