In the diagram, OABCDEFG is a cuboid in which OA = 3 units, OC = 2 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OD and OC respectively. M is the midpoint of EF. The position vector of P is i + j + 2k. [Figure 11]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [B1] (a) Obtain 3i + 2j + k Accept coordinates in place of position vector.
2. [B1] (b) AM or AP correct soi AM = 2j + k, or AP = −2i + j + 2k.
3. [M1] (b) Carry out correct process for evaluating the scalar product of AM and AP or MA and PA: 0+2+2.
4. [M1] (b) Using the correct process for the moduli, divide the scalar product by the product of the moduli and obtain the inverse cosine of the result For their vectors. θ = cos⁻¹(4 / 3√5).
5. [A1] (b) Obtain answer 53.4° or 0.932°
6. [B1FT] (c) Find PQ (or QP) for a general point Q on the line passing through O and M, e.g. PQ = - (i + j + 2k) + μ(3i + 2j + k). Follow their M.
7. [M1] (c) Calculate the scalar product of PQ and a direction vector for the line passing through O and M and equate to zero
8. [A1] (c) Solve and obtain correct solution e.g. μ = -1/2
9. [DM1] (c) Carry out method to calculate PQ √.5² +0+1.5².
10. [A1] (c) Obtain answer √10 / 2 Or exact equivalent.
11. [B1] (c) Alternative Method 1: Find PQ (or QP) for a general point Q on the line passing through O and M, e.g. PQ = - (i + j + 2k) + μ(3i + 2j + k). Follow their M.
12. [M1] (c) Alternative Method 1: Use a correct method to express PQ² (or PQ) in terms of μ
13. [A1] (c) Alternative Method 1: Obtain a correct equation in any form e.g. PQ² = (1 + 3μ)² + (1 + 2μ)² + (2 + μ)²
14. [DM1] (c) Alternative Method 1: Carry out a complete method for finding its minimum e.g. 6(1+3μ)+4(1+2μ) + 2(2+ μ) = 0, μ=−1/2
15. [A1] (c) Alternative Method 1: Obtain answer √10 / 2 Or exact equivalent.
16. [B1] (c) Alternative Method 2: Calling (0, 0, 0) A, state PA (or AP) in component form, e.g. i + j + 2k
17. [M1] (c) Alternative Method 2: Use a scalar product to find the projection of PA (or AP) on the line passing through O and M
18. [A1OE] (c) Alternative Method 2: Obtain correct answer 7/√14
19. [M1] (c) Alternative Method 2: Use Pythagoras to find the perpendicular d = √(AP² - AQ²) = √(1²+1²+2² - (7/√14)²)
20. [A1] (c) Alternative Method 2: Obtain answer √10 / 2 Or exact equivalent.
21. [B1] (c) Alternative Method 3: Calling (0, 0, 0) A, state PA (or AP) in component form, e.g. i + j + 2k
22. [M1] (c) Alternative Method 3: Calculate the vector product of PA and a direction vector for the line passing through O and M
23. [A1] (c) Alternative Method 3: Obtain correct answer, e.g. 3i - 5j + k
24. [M1] (c) Alternative Method 3: Divide modulus of the product by that of the direction vector e.g. √(3²+5²+1²) / √(3²+2²+1²).
25. [A1] (c) Alternative Method 3: Obtain answer √10 / 2 Or exact equivalent.