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A-LevelMathematicsVectors9709/12 · Oct/Nov 2012 · Q78 Marks

The position vectors of the points A and B, relative to an origin O, are given by OA = (1, 0, 2) and OB = (k, -k, 2k), where k is a constant.

  1. (i)In the case where k = 2, calculate angle AOB.[4]
  2. (ii)Find the values of k for which AB is a unit vector.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2012). The mark or mark type is shown in brackets.

  1. [M1] Use of x₁x₂ + y₁y₂ + z₁z₂ (dot product)
  2. [M1] Product of 2 moduli (magnitudes)
  3. [M1] All connected correctly (using dot product formula for angle)
  4. [A1] → θ = 24.1°
  5. [M1] AB = (k-1, -k, 2k-2) (or BA)
  6. [M1] (k-1)² + k² + (2k-2)² (sum of squares for magnitude)
  7. [M1] → 6k² - 10k + 4 = 0 (correct quadratic)
  8. [A1] → k = 1 or ²⁄₃

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About this question

Topic
Vectors, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, Oct/Nov 2012 Paper 1 Variant 2, question 7.

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