With respect to the origin O, the points A, B, C and D have position vectors given by OA = (3, -1, 2), OB = (1, 2, -3), OC = (-2, 5, 1) and OD = (-6, 11, 1).

Mark scheme

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

1. [M1] Carry out correct process for evaluating the scalar product of OA and OB
2. [A1] Using the correct process for the moduli, divide the scalar product by the product of the moduli and obtain cos⁻¹{±(3 – 2 – 6)/[√(3² + (−1)² +2²)√(1² + 2²+(-3)²)]}
3. [A1] Obtain answer 110.9° or 1.94°
4. [M1] Use a correct method to form an equation for line through AB
5. [A1] Obtain r = 3i - j + 2k + μ₁(2i – 3j + 5k) OE e.g. r = i + 2j – 3k + μ₂(−2i + 3j – 5k). Need r or (x, y, z).
6. [B1] Obtain a correct equation for line through CD e.g. [r = ] i-2j + 5k + λ₁(-4i + 4j – 6k) OE e.g. [r = ] 5i – 6j + 11k + λ₂(–4i + 4j – 6k). r can be omitted or another symbol used.
7. [M1] Equate two pairs of components of general points on their l and their CD and solve for λ or for μ
8. [A1] Obtain e.g. λ₁ = −2 or μ₁ = 3 or λ₂ = −1 or μ₂ = −4
9. [A1] Obtain position vector 9i – 10j + 17k Condone (9, –10, 17) but not (9i, – 10j, 17k).