The equations of two straight lines l₁ and l₂ are l₁: r = i-2j+3k + λ(2i-j+ak) and l₂: r = -i-j-k + μ(3i-2j-2k), where a is a constant. The lines l₁ and l₂ are perpendicular.

Mark scheme

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

1. [B1] Carry out correct process for evaluating the scalar product of direction vectors, equate the result to zero and obtain given value of a = 4
2. [B1] Express general point of at least one line correctly in component form, i.e. (1 + 2λ, −2 – λ, 3 + 4λ) or (−1 + 3μ, −1 – 2μ, −1 – 2μ)
3. [M1] Equate at least two pairs of corresponding components and solve for λ or for μ
4. [A1] Obtain λ = −1 or μ = 0
5. [A1] Obtain position vector of point of intersection is – i-j-k
6. [M1] Equate one component of l₁ to matching component of A and solve to find λ
7. [A1] Use λ = −3 in equation of l₁ and show this gives position vector of A
8. [M1] Method to find position vector of B
9. [A1] Obtain position vector of B is 3i – 3j + 7k