A-LevelMathematicsVectors9709/12 · Oct/Nov 2014 · Q77 Marks
The diagram shows a pyramid OABCX. The horizontal square base OABC has side 8 units and the centre of the base is D. The top of the pyramid, X, is vertically above D and XD = 10 units. The mid-point of OX is M. The unit vectors i and j are parallel to OA and OC respectively and the unit vector k is vertically upwards. [Figure of a pyramid OABCX with vectors i, j, k and labels M, D, 10, 8]
- (i)Express the vectors AM and AC in terms of i, j and k.[3]
- (ii)Use a scalar product to find angle MAC.[4]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.
- [2] AM = -6i + 2j + 5k. co –1 each error
- [1] AC = -8i + 8j. co
- [1] AM.AC = (-6)(-8) + (2)(8) + (5)(0) = 48 + 16 = 64. Use of x₁x₂ + etc. with suitable vectors
- [2] 64 = |AM||AC|cosθ = √((-6)²+2²+5²)√(8²+8²)cosθ = √65√128cosθ. Product of moduli. Correct link.
- [1] → θ = 45.4°. co
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