Question

A plane flies 452 miles north and

then 767 miles west.
What is the direction of the
plane's resultant vector?
Hint: Draw a vector diagram.
Ө 0 = [ ? ]°
Round your answer to the nearest hundredth.

Answer 1

149.49° (nearest hundredth)

Explanation:

To calculate the direction of the plane's resultant vector, we can draw a vector diagram (see attachment).

• The starting point of the plane is the origin (0, 0).
• Given the plane flies 452 miles north, draw a vector from the origin north along the y-axis and label it 452 miles.
• As the plane then flies 767 miles west, draw a vector from the terminal point of the previous vector in the west direction (to the left) and label it 767 miles.

Since the two vectors form a right angle, we can use the tangent trigonometric ratio.

`\boxed{\begin{minipage}{7 cm}\underline{Tangent trigonometric ratio} \\\\$ \tan x=(O)/(A)$\\\\where:\\ \phantom{ww}$\bullet$ $x$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\end{minipage}}`

The resultant vector is in quadrant II, since the plane is travelling north (positive y-direction) and then west (negative x-direction).

As the direction of a resultant vector is measured in an anticlockwise direction from the positive x-axis, we need to add 90° to the angle found using the tan ratio.

The angle between the y-axis and the resultant vector can be found using tan x = 767 / 452. Therefore, the expression for the direction of the resultant vector θ is:

`\theta=90^(\circ)+\arctan \left((767)/(452)\right)`

`\theta=90^(\circ)+59.4887724...^(\circ)`

`\theta=149.49^(\circ)\; \sf (nearest\;hundredth)`

Therefore, the direction of the plane's resultant vector is approximately 149.49° (measured anticlockwise from the positive x-axis).

This can also be expressed as N 59.49° W.