SPM.2.AHL.TZ0.7

pestleMathematicsAAHLPaper 2SPM· ahl-3-14-vector-equation-of-linesource ↗

Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time t hours after midday are given by

r_A = $\left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 5 \\ 8 \end{array}} \right)$

r_B = $\left( {\begin{array}{*{20}{c}} 7 \\ { - 3} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 0 \\ {12} \end{array}} \right)$

where distances are measured in kilometres.

Find the minimum distance between the two ships.

Markscheme / solution

attempting to find r_B − r_A for example (M1)

r_B − r_A = $\left( {\begin{array}{*{20}{c}} 3 \\ { - 6} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 5} \\ 4 \end{array}} \right)$

attempting to find |r_B − r_A| M1

distance $d\left( t \right) = \sqrt {{{\left( {3 - 5t} \right)}^2} + {{\left( {4t - 6} \right)}^2}} \left( { = \sqrt {41{t^2} - 78t + 45} } \right)$ A1

using a graph to find the $d$ − coordinate of the local minimum M1

the minimum distance between the ships is 2.81 (km) $\left( { = \frac{{11\sqrt {41} }}{{41}}\,\left( {{\text{km}}} \right)} \right)$ A1

[5 marks]

Examiners’ report

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