With respect to the origin O, the points A, B and C have position vectors given by OA = i+2j, OB = i+3j-2k and OC = 2i-j+3k. The line l passes through B and C.

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 the line through B and C. E.g. r=OB + λBC.
2. [A1] Obtain r = i +3j-2k + λ(i−4j+5k). OE. Must have r or component or column vector form. E.g. r = 2i - j+3k + λ(−i + 4j−5k). λ=... scores A0.
3. [B1] Find AP for a general point P on l. Allow unsimplified. E.g. j− 2k + λ(i−4j+5k) or i−3j+3k + μ(i −4j+5k).
4. [M1] Calculate the scalar product of AP (not OP) and a direction vector for l and equate the result to zero. E.g. (j-2k+λ(i−4j+5k)). (i-4j+5k) = 0 => λ−4(1-4λ)+5(−2+5λ) = 0.
5. [A1] Obtain λ = 1/2 or μ = -1/2. Or correct equivalents.
6. [A1] Obtain i+j+1/2k. Or equivalent column vector.
7. [M1] Use a correct method to find their position vector of D. E.g. OD=OA+2AP. Allow a slip in one component.
8. [A1] Obtain i+j+1/2k. Or equivalent column vector.