With respect to the origin O, the points A and B have position vectors given by OA = (1, 2, 1) and OB = (3, 1, -2). The line l has equation r = (2, 3, 1) + λ(1, -2, 1).

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 = (2, -1, -3)
2. [M1] Use the correct process to calculate the scalar product of a pair of relevant vectors, e.g. their AB and a direction vector for l
3. [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
4. [A1] Obtain answer 83.7° or 1.46 radians
5. [B1] State or imply ± AP and ± BP in component form, i.e. (1 + λ, 1 – 2λ, λ) and (− 1 + λ, 2 – 2λ, 3 + λ), or equivalent
6. [*M1] Form an equation in λ by equating moduli or by using cos BAP = cos ABP
7. [A1] Obtain a correct equation in any form
8. [DM1] Solve for λ and obtain position vector
9. [A1] Obtain correct position vector for P in any form, e.g. (8, – 9, 7) or 8i – 9j + 7k