IB Maths AI HL AI 3.3 Vectors and matrix transformations questions

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Question 1

Paper 1Medium6 marks

Two survey boats, A and B, move at constant velocities in a horizontal plane. Coordinates are measured in metres, with the xx-axis pointing east and the yy-axis pointing north. At t=0t=0 their position vectors are

rA=(120,30),rB=(20,100),\mathbf r_A=(120,30), \qquad \mathbf r_B=(20,100),

and their velocity vectors are

vA=(3,1),vB=(5,−2),\mathbf v_A=(3,1), \qquad \mathbf v_B=(5,-2),

in m s−1\text{m s}^{-1} The time tt is measured in seconds.

(a)

Find an expression, in terms of tt for the relative position vector of B from A, AB→\overrightarrow{AB}

[2]
(b)

Using your answer to part (a), determine the time t≥0t\geq 0 at which the boats are closest and their minimum separation. Give both answers to 3 significant figures.

[4]

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