Relative to the origin O, the points A, B and C have position vectors given by OA = (2 3 3), OB = (4 1 2) and OC = (3 -2 -4). The quadrilateral ABCD is a parallelogram.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [B1] Obtain a vector for one side of the parallelogram e.g. AB = (2, -2, -1) or BC = (-1, -3, -6).
2. [M1] Correct method to obtain ±OD e.g. OD = OA + BC. MO if use AB = CD or BC = DA.
3. [A1] Obtain OD = i - 4j - 3k Any equivalent form. Accept coordinates.
4. [M1] Using the correct process, evaluate the scalar product BA.BC (2+10-6) Scalar product of two relevant vectors. OE
5. [M1] Using the correct process for the moduli, divide the scalar product by the product of the moduli. (2+10-6) / (√9×√62)
6. [A1] Obtain answer 2/√62 ISW Or simplified equivalent i.e. √62 / 31
7. [B1 FT] State or imply sin θ = √58/√62 Follow their cos θ.
8. [M1] Use correct method to find the area of ABCD e.g. 2 × ½ BA × BC sin θ. Condone decimals.
9. [A1 FT] Correct unsimplified expression for the area e.g. 2 × ½ × 3 × √62 × sin θ. Condone decimals. Follow their sides and angle.
10. [A1] Obtain answer 3√58 Correct only.