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Vectors and matrices — Question 15

QCAA 2024, Paper 2 · 8 marks

Q15 · 2024 · Technology-activeSimple familiar8 marks

QUESTION 15 (8 marks)

The vectors representing the position (m) of particles A and B are given by rA=(4t−9)i^−2(5−t)j^−8k^\mathbf r_A=(4t-9)\hat{\mathbf i}-2(5-t)\hat{\mathbf j}-8\hat{\mathbf k} and rB=(t2+1)i^−3j^+(4−at2)k^\mathbf r_B=(t^2+1)\hat{\mathbf i}-3\hat{\mathbf j}+(4-at^2)\hat{\mathbf k} respectively, where tt is the time of motion for 0≤t≤100\le t\le10 seconds.
a)
Show that particle A passes through the point P(5,−3,−8)P(5,-3,-8).
[2 marks]
b)
Given that particle B also passes through point PP, determine the value of aa.
[2 marks]
c)
Determine the vector that represents the displacement of particle B relative to particle A during the given time of motion. Express your answer in simplest form.
[1 mark]
d)
Use your result from Question 15c) to determine the shortest distance between particles A and B during the given time of motion.
[3 marks]
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