With respect to the origin O, the position vectors of the points A, B, C and D are given by OA = (2, 1, 5), OB = (4, -1, 1), OC = (1, 1, 2) and OD = (2, 3, 3).

Mark scheme

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

1. [B1] Obtain AB = (2, -2, -4) and CD = (1, 2, 1) [incorrect OCR]
2. [M1] Use the correct process for calculating the modulus of both vectors to obtain AB and CD
3. [A1] Using exact values, verify that AB = 2CD
4. [M1] Use the correct process to calculate the scalar product of the relevant vectors (their AB and CD)
5. [M1] Divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result
6. [A1] Obtain answer 99.6° (or 1.74 radians) or better
7. [B1] State correct vector equations for AB and CD in any form, e.g. (r =)(2,1,5) + λ(2,-2,-4) and (r =)(1,1,2) + μ(1,2,1)
8. [M1] Equate at least two pairs of components of their lines and solve for λ or for μ
9. [A1] Obtain correct pair of values from correct equations
10. [A1] Verify that all three equations are not satisfied and that the lines do not intersect