Two planes p and q have equations x + y + 3z = 8 and 2x – 2y + z = 3 respectively.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

1. [1] State or imply a correct normal vector to either plane, e.g. i + j + 3k or 2i – 2j + k
2. [1] Carry out correct process for evaluating the scalar product of two normal vectors
3. [1] Using the correct process for the moduli, divide the scalar product of the two normals by the product of their moduli and evaluate the inverse cosine of the result
4. [1] Obtain final answer 72.5° or 1.26 radians
5. [1] Substitute y = 2 in both plane equations and solve for x or for z
6. [1] Obtain x = 3 and z = 1
7. [1] EITHER: Use scalar product to obtain an equation in a, b and c, e.g. a + b + 3c = 0
8. [1] Form a second relevant equation, e.g. 2a - 2b + c = 0, and solve for one ratio, e.g. a : b
9. [1] Obtain final answer a:b:c=7:5:-4
10. [1] OR1: Calculate the vector product of relevant vectors, e.g. (i+j+3k)×(2i – 2j+k)
11. [1] Obtain two correct components (for OR1)
12. [1] Obtain correct answer, e.g. 7i+5j-4k (for OR1)
13. [1] OR2: Using relevant vectors, form a two-parameter equation for the plane
14. [1] State a correct equation, e.g. r = 3i + 2j + k + λ(i + j + 3k) + α(2i−2j+k)
15. [1] State 3 correct equations in x, y, z, λ and α
16. [1] Eliminate λ and α
17. [1] OR3: Use the direction vector of the line of intersection of the two planes as normal vector to the plane
18. [1] Two correct components (for OR3)
19. [1] Three correct components (for OR3)
20. [1] Use coordinates of A and values of a, b and c in general equation and find d
21. [1] Obtain answer 7x + 5y – 4z = 27, or equivalent