With respect to the origin O, the vertices of a triangle ABC have position vectors OA = 2i + 5k, OB = 3i + 2j + 3k and OC = i + j + k.

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

1. [B1] (a) State AB(or BA) and BC(or CB) in vector form
2. [M1] (a) Calculate their scalar product
3. [A1] (a) Show product is zero and confirm angle ABC is a right angle
4. [M1] (b) Use correct method to calculate the lengths of AB and BC
5. [A1] (b) Show that AB = BC and the triangle is isosceles
6. [B1] (c) State a correct equation for the line through B and C, e.g. r = i + j + k + λ(2i + j + 2k) or r = 3i + 2j + 3k + μ(−2i − j − 2k)
7. [M1] (c) Taking a general point of BC to be P, form an equation in λ by either equating the scalar product of OP and BC to zero, or applying Pythagoras to triangle OBP (or OCP), or setting the derivative of |OP| to zero
8. [A1] (c) Solve and obtain λ = -5/9
9. [A1] (c) Obtain answer (1/3)√2, or equivalent