With respect to the origin O, the points A and B have position vectors 2i+4k and 5i+j+6k respectively. The line l₁ passes through the points A and B.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [M1] Use a correct method to form an equation for l₁ Accept column vectors.
2. [A1] Obtain r = (2i + 4k) + λ(3i + j + 2k) OE, e.g. r = (5i + j + 6k) + λ(3i + j +2k). Must have r = ..., or in x, y, z or R = ....
3. [B1FT] Express general point of a line in component form E.g. (2+3λ, λ, 4+2λ) or (2+μ, 1+2μ, 5+3μ)
4. [M1] Equate two pairs of components of l₁ and their l₂, and solve for λ or μ
5. [A1] Obtain e.g. λ = -1/3, μ = -1/3 Or λ = -1, μ = -1.
6. [A1] Show that this does not fit the third component and hence the lines do not intersect. 4 + 2(-1/3) ≠ 5 + 3(-1/3) or 16/3 ≠ 4
7. [M1] Carry out the correct process for evaluating the scalar product of the direction vector of l₁ and l₂ E.g. (i+2j+3k)·(3i+j+2k)= 3+2+6.
8. [M1] Using the correct process for the moduli, divide their scalar product by the product of the moduli of their vectors and evaluate the inverse cosine of the result
9. [A1] Obtain AWRT 38.2° or 0.667 radians