The points A and B have position vectors 2i + j + k and i – 2j + 2k respectively. The line l has vector equation r = i + 2j – 3k + μ(i – 3j – 2k).

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2022). The mark or mark type is shown in brackets.

1. [1] Obtain direction vector –i – 3j + k
2. [1] Use a correct method to form a vector equation
3. [1] Obtain answer r = 2i + j + k + λ(–i – 3j + k) or r = i – 2j + 2k + λ(–i – 3j + k)
4. [1] Carry out the correct process for evaluating the scalar product of the direction vectors.
5. [1] Using the correct process for the moduli, divide the scalar product by the product of the moduli and find the inverse cosine of the result for any 2 vectors
6. [1] Obtain answer 61.1°
7. [1] Express general point of AB or l in component form, e.g. (2 – λ, 1 – 3λ, 1 + λ) or (1 + μ, 2 – 3μ, – 3 – 2μ)
8. [1] Equate at least two pairs of components and solve for λ or for μ
9. [1] Obtain a correct answer for λ or μ, e.g. λ = 6, 1/4, or –2/3; μ = –5, 2/3 or –11/9
10. [1] Verify that all three equations are not satisfied, and the lines do not intersect