Given that P is 25 m on a bearing of 330º from Q, how far south of P is Q?

Analyze the question

P is 25 m from Q on a bearing of 330°. We need how far Q is south of P, which is the southward component of the displacement from P to Q. Equivalently, it is the northward component of the displacement from Q to P.

Concept

A bearing is measured clockwise from North. A bearing of 330° is 30° west of North (360° − 330° = 30°). So the line QP makes an angle of 30° with the north-south line through Q.

Solution

1. Draw Q, then draw P at a distance of 25 m in the direction N30°W from Q.
2. Drop a perpendicular from P to the north line through Q. This gives a right-angled triangle where: • Hypotenuse QP = 25 m • The angle at Q between the north line and QP = 30°
3. The northward distance of P from Q (the distance Q is south of P) is the side adjacent to the 30° angle:

North component = 25 cos 30° = 25 × 0.8660 = 21.65 m ≈ 21.7 m

(The westward component would be 25 sin 30° = 12.5 m, which is not what is asked.)

Answer: Q is about 21.7 m south of P, which is option B.

Tip: For bearing problems, first convert the bearing to an acute angle from the North or South line. The component along the north-south line uses cos, and the component along the east-west line uses sin. Always draw a quick sketch to check which angle you are using.