Relative to the origin O, the points A, B and D have position vectors given by OA = i + 2j + k, OB = 2i + 5j + 3k and OD = 3i + 2k. A fourth point C is such that ABCD is a parallelogram.

Mark scheme

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

1. [B1] State or imply AB or AD in component form
2. [M1] Use a correct method for finding the position vector of C
3. [A1] Obtain answer 4i + 3j + 4k, or equivalent
4. [M1] Using the correct process for the moduli, compare lengths of a pair of adjacent sides, e.g. AB and AD
5. [A1] Show that ABCD has a pair of unequal adjacent sides
6. [B1] State or imply AB or AD in component form
7. [M1] Use a correct method for finding the position vector of C
8. [A1] Obtain answer 4i + 3j + 4k, or equivalent
9. [M1] Use the correct process to calculate the scalar product of AC and BD, or equivalent
10. [A1] Show that the diagonals of ABCD are not perpendicular
11. [M1] Use the correct process to calculate the scalar product of a pair of relevant vectors, e.g. AB and AD
12. [M1] Using the correct process for the moduli, divide the scalar product by the product of the moduli of the two vectors and evaluate the inverse cosine of the result
13. [A1] Obtain answer 100.3°
14. [M1] Use a correct method to calculate the area, e.g. calculate AB.AC sin BAD
15. [A1 FT] Obtain answer 11.0 (FT on angle BAD)