Two lines l and m have equations r = 3i + 2j + 5k + s(4i – j + 3k) and r = i − j − 2k + t(−i + 2j + 2k) respectively.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [M1] Use correct method to evaluate the scalar product of relevant vectors
2. [A1] Obtain answer zero and deduce the given statement
3. [B1] Express general point of `l` or `m` in component form, e.g. `(3 + 4s, 2 – s, 5 + 3s)` or `(1 – t, – 1 + 2t, – 2 + 2t)`
4. [M1] Equate at least two pairs of components and solve for s or for t
5. [A1] Obtain correct answer `s = -1` and `t = 2`
6. [A1] Verify that all three equations are satisfied
7. [A1] State position vector of the intersection `–i + 3j +2k`, or equivalent
8. [*M1] Taking a general point P on m, form an equation in t by either equating a relevant scalar product to zero, or equating the derivative of `|OP|²` to zero, or taking a specific point Q on m, e.g. `(1, –1, –2)`, using Pythagoras in triangle OPQ
9. [A1] Obtain `t = 7/9`
10. [DM1] Carry out correct method to find `|OP|`
11. [A1] Obtain `(4/3)√5`