With respect to the origin O, the points A, B and C have position vectors given by OA = (2, 1, -3), OB = (0, 4, 1), and OC = (-3, -2, 2).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [M1] Use a correct method to find OD
2. [A1] Obtain position vector of D is 3i – 11j – 10k
3. [*M1] Carry out correct method for finding a vector equation for AC or BD
4. [A1ft] Both diagonal equations correct.
5. [DM1] Equate at least two pairs of corresponding components and solve for λ or for μ
6. [A1] Obtain λ = -1/4 or μ = 1/4
7. [A1] Obtain position vector of P is ¾ i + ¼ j - ⁷⁄₄ k
8. [B1ft] Find direction vector BA = 2i – 3j – 4k and BC = -3i – 6j + k or equivalent
9. [M1] Carry out correct process for evaluating the scalar product of two relevant vectors
10. [M1] Using the correct process for the moduli, divide their scalar product by the product of their moduli and evaluate the inverse cosine of the result to obtain an angle
11. [A1] Obtain answer 77.3° (or 1.35 radians)