Specialist Maths — Question 14
QCAA 2023, Paper 1 · 6 marks
Q14 · 2023 · Technology-freeSimple familiar6 marks
QUESTION 14 (6 marks)
Consider a cube with three edges positioned along the x-, y- and z-axes on the Cartesian plane as shown. Points O, A and B are vertices of the cube. 
a)Given OA=2i^, determine OB. Express your answer in terms of j^ and k^. b)Calculate OA×OB. c)Consider the triangle formed by joining points O, A and B. Use the result from Question 14b) to determine the area of the triangle.
d)Let points M and N be the midpoints of the triangle’s sides OA and OB respectively. Determine MN. e)Use the result from Question 14d) to show that the length of AB is twice the length of MN.
WORKED SOLUTION
QCAA guide · typeset solution
6 marksANSWEROB=2j^+2k^; area 22 units2; MN=−i^+j^+k^. Worked solution
a) OB=2j^+2k^. b) OA×OB=2i^×(2j^+2k^)=−4j^+4k^. c) The area is 21∣−4j^+4k^∣=22 units2. d) M=i^ and N=j^+k^, so MN=−i^+j^+k^. e) AB=−2i^+2j^+2k^=2MN, hence ∣AB∣=2∣MN∣. correctly determines OB calculates OA×OB states an expression representing the area of triangle OAB
calculates the area of triangle OAB
determines MN shows that AB=2MN One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.
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