With respect to the origin O, the point A has position vector given by OA = i + 5j + 6k. The line l has vector equation r = 4i + k + λ(−i + 2j + 3k).

Mark scheme

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

1. [M1] Using the correct process find the scalar product of direction vectors of l and OA
2. [M1] Using the correct process for the moduli, divide the scalar product by the product of the moduli and find the inverse cosine of the result
3. [A1] Obtain answer 23.6°.
4. [B1] Taking a general point P on l, state AP (or PA) in component form, e.g. (3 – λ, – 5 + 2λ, – 5 + 3λ)
5. [M1] Either equate scalar product of AP and direction vector of l to zero and solve for λ or use Pythagoras in a relevant triangle and solve for λ
6. [A1] Obtain λ = 2
7. [A1] State that the position vector OP* of the foot is 2i + 4j + 7k
8. [M1] Set up a correct method for finding the position vector of the reflection of A in l
9. [A1] Obtain answer 3i + 3j + 8k or (3, 3, 8)