Two planes have equations 2x + 3y – z = 1 and x – 2y + z = 3.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.

1. [B1] State or imply a correct normal vector to either plane, e.g. 2i + 3j – k, or i – 2j + k
2. [M1] Carry out correct process for evaluating the scalar product of two normal vectors
3. [M1] Using the correct process for the moduli, divide the scalar product of the two normal vectors by the product of their moduli and evaluate the inverse cosine of the result
4. [A1] Obtain answer 56.9° or 0.994 radians
5. [M1] Carry out a complete strategy for finding a point on the line (call the line l)
6. [A1] Obtain such a point, e.g. (1, 1, 4)
7. [B1] State a correct equation for a direction vector ai + bj + ck for l, e.g. 2a + 3b - c = 0
8. [M1] State a second equation, e.g. a - 2b + c = 0, and solve for one ratio, e.g. a : b
9. [A1] Obtain a : b : c = 1 : -3 : -7, or equivalent
10. [A1] State a correct answer, e.g. r = i + j + 4k + λ(i – 3j – 7k)