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Relative to a point
O, the position vector of a ship is
(12 i+16 j)km. If the ship sails with a constant speed of
24km//h on a bearing of
060^(@), find its position vector after 4 hours. Give your answer correct to 3 significant figures.

Relative to a point $O$ , the position vector of a ship is $(12 i+16 j) \mathrm{km}$ . If the ship sails with a constant speed of $24 \mathrm{~km} / \mathrm{h}$ on a bearing of $060^{\circ}$ , find its position vector after $4$ hours. Give your answer correct to $3$ significant figures.

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Find the magnitude and direction of a vector sum

Full solution

Q. Relative to a point

O

, the position vector of a ship is

(12 i+16 j) \mathrm{km}

. If the ship sails with a constant speed of

24 \mathrm{~km} / \mathrm{h}

on a bearing of

060^{\circ}

, find its position vector after

4

hours. Give your answer correct to

3

significant figures.

Calculate displacement vector: Calculate the displacement vector after $4$ hours. $\newline$ The ship travels at $24\,\text{km/h}$ and sails for $4$ hours. Distance $=$ speed $\times$ time $= 24\,\text{km/h} \times 4\,\text{h} = 96\,\text{km}$ .
Convert bearing to direction angle: Convert the bearing to a direction angle. $\newline$ A bearing of $060°$ corresponds to $60°$ north of east. This is $30°$ from the positive x-axis (east direction).
Calculate components using trigonometry: Calculate the components of the displacement vector using trigonometry. $\newline$ Using the direction angle: $\newline$ $x$ -component $= 96 \, \text{km} \times \cos(30°) = 96 \times 0.866 = 83.136 \, \text{km}$ , $\newline$ $y$ -component $= 96 \, \text{km} \times \sin(30°) = 96 \times 0.5 = 48 \, \text{km}$ .
Add initial position vector: Add the initial position vector to the displacement vector. $\newline$ Initial position vector = $(12i + 16j)$ km, $\newline$ Displacement vector = $(83.136i + 48j)$ km, $\newline$ New position vector = $(12 + 83.136)i + (16 + 48)j = (95.136i + 64j)$ km.

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