Question 1

Paper 2Medium6 marks

For the first 10 hours of a survey, two vessels move in a horizontal plane at constant velocity. Their position vectors, relative to a fixed origin, are

rA(t)=(1.5, 2.0)+t(0.40, 0.30),rB(t)=(6.7, 5.1)+t(−0.35, −0.10),\mathbf r_A(t)=(1.5,\,2.0)+t(0.40,\,0.30),\qquad \mathbf r_B(t)=(6.7,\,5.1)+t(-0.35,\,-0.10),

where coordinates are in kilometres and tt is the time in hours after 08:00.

(a)

Write down the initial position vector of B relative to A, AB→(0)\overrightarrow{AB}(0)

[1]
(b)

Show that

AB→(t)=(5.2, 3.1)+t(−0.75, −0.40).\overrightarrow{AB}(t)=(5.2,\,3.1)+t(-0.75,\,-0.40).
[2]
(c)

Hence determine the time, tt and the distance between the vessels when they are closest during the 10-hour survey. Give both answers to 3 significant figures.

[3]

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