Two lines l and m have equations r = 2i – j + k + s(2i + 3j – k) and r = i + 3j + 4k + t(i + 2j + k) respectively.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. [M1] Equate at least two pairs of components and solve for s or for t
2. [A1] Obtain correct answer for s or t, e.g. s = – 6, t = – 11
3. [A1] Verify that all three equations are not satisfied and the lines fail to intersect
4. [B1] State that the lines are not parallel
5. [B1] EITHER: Use scalar product to obtain a relevant equation in a, b and c, e.g. 2a + 3b − c = 0
6. [M1] Obtain a second equation, e.g. a + 2b + c = 0, and solve for one ratio, e.g. a : b
7. [A1] Obtain a:b:c and state correct answer, e.g. 5i – 3j + k, or equivalent
8. OR: Attempt to calculate vector product of relevant vectors, e.g. (2i + 3j – k)×(i + 2j + k)
9. [M1] Obtain two correct components
10. [A1] Obtain correct answer, e.g. 5i – 3j + k
11. [B1] EITHER: State position vector or coordinates of the mid-point of a line segment joining points on l and m, e.g. 3/2 i + 1/2 j + 5/2 k
12. [M1] Use the result of (ii) and the mid-point to find d
13. [A1] Obtain answer 5x – 3y + z = 7, or equivalent
14. OR: Using the result of part (ii), form an equation in d by equating perpendicular distances to the plane of a point on l and a point on m
15. [M1] State a correct equation, e.g. |(14-d)/√35| = |-d/√35|
16. [A1] Solve for d and obtain answer 5x – 3y + z = 7, or equivalent