O-LevelAdditional MathematicsVectors in two dimensions4037/21 · May/June 2022 · Q87 Marks

In this question, i is a unit vector due east and j is a unit vector due north. Distances are measured in kilometres and time is measured in hours. At 0900, ship A leaves a point P with position vector 5i+16j relative to an origin O. It sails with a constant speed of 6√3 on a bearing of 120°.

(a)Show that the velocity vector of A is 9i-3√3j.[2]
(b)Find the position vector of A at 1200.[1]
(c)At 1100 ship B leaves a point Q with position vector 29i+16j. It sails with constant velocity -12√3j. Write down the position vector of B, t hours after it starts sailing.[1]
(d)Find the distance between the two ships at 1200.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[B2] x = 6√3sin60 y = -6√3 cos 60 oe and completion to 9i – 3√3j
[B1] for either x or y correct
[SC1] for verification that 9i - 3√3j has a bearing of 120° and that 9i – 3√3j has a magnitude of 6√3
[B1] (5i+16j)+3(9i-3√3j) oe, isw
[B1] 29i+16j+t(-12√3j) oe, isw
[B1] Forms AB or BA when t = 1 e.g. BA = (32i+(16-9√3)j) –(29i+(16-12√3)j) oe FT their (b) and (c) with t=1
[M1] √3² + (3√3)² FT their AB or BA
[A1] 6 (km) cao

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