Relative to the origin O, the points A, B and C have position vectors given by OA = (1, 1, 1), OB = (3, 1, 2) and OC = (5, -2, -2).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [B1] (a) State or imply AB or AC correctly in component form
2. [M1] (a) Using the correct process with relevant vectors to evaluate the scalar product AB.AC
3. [M1] (a) Using the correct process for the moduli, divide their scalar product by the product of their moduli to obtain cosθ or 1/3
4. [A1] (a) Obtain answer 1/3
5. [M1] (b) Use correct method to find an exact value for the sine of angle BAC from their (a)
6. [A1] (b) Obtain answer 2√2/3, or equivalent
7. [M1] (b) Use correct area formula to find the area of triangle ABC with their versions of relevant vectors
8. [A1] (b) Obtain answer 5√2 or √50
9. [M1] (Alternative method 1 for 6b) Use correct method to find the perpendicular distance from A to BC (or B to AC or C to AB)
10. [A1] (Alternative method 1 for 6b) Obtain √75
11. [M1] (Alternative method 1 for 6b) Use correct area formula to find the area of triangle ABC
12. [A1] (Alternative method 1 for 6b) Obtain answer 5√2 or √50
13. [M1] (Alternative method 2 for 6b) Correct method to find the semi-perimeter
14. [A1] (Alternative method 2 for 6b) Obtain 4 + √6
15. [M1] (Alternative method 2 for 6b) Correct application of Hero's (Heron's) formula
16. [A1] (Alternative method 2 for 6b) Obtain answer 5√2 or √50