A-LevelMathematicsVectors9709/32 · Feb/Mar 2024 · Q98 Marks

Relative to the origin O, the position vectors of the points A, B and C are given by OA = 5i-2j+k, OB = 8i+2j-6k and OC = 3i+4j-7k.

(a)Show that OABC is a rectangle.[4]
(b)Use a scalar product to find the acute angle between the diagonals of OABC.[4]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

[M1] Find the scalar product of a pair of adjacent sides
[A1] Show that the sides are perpendicular
[M1] Compare a pair of opposite sides
[A1] Show that they are parallel and equal in length and hence OABC is a rectangle
[B1] AC = ±(-2i+6j-8k) or AC/2 = ±(-li+3j-4k)
[M1] Scalar product of a pair of relevant vectors
[M1] 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.
[A1] Obtain answer 65.(0)°

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Topic: Vectors, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, Feb/Mar 2024 Paper 3 Variant 2, question 9.

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