Relative to the origin O, the position vectors of the points A, B and C are given by `OA = (2 5)`, `OB = (4 3)`, `OC = (10 6)`. The point D is such that ABCD is a parallelogram.

Mark scheme

Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.

1. [6] (i) BA.BC or AB.CB; B1 Correct two vectors for angle ABC.; BA = (-2 2); BC = (6 -3); M1 Correct method for one of the sides.; BA.BC = -8; M1 Correct use for any pair of vectors.; |BA| = √((-2)²+2²) = √8; |BC| = √(6²+(-3)²) = √45; M1 Correct method for moduli.; cos θ = (-8) / (√8 * √45); M1 A1 All linked correctly. co (67.6° usually gets 4/6); → θ = 112.4° or 1.96 radians
2. [2] (ii) OD = OA + AD = OA + BC; M1 Correct method. (allow for d = a + b − c or for d = a + c − b or for d = b + c − a); A1√ Al for his BC.; → OD = (8 8)