2021 HSC Mathematics Extension 1
Question 14(a):
Vectors
A plane needs to travel to a destination that is on a bearing of . The engine is set to fly at a constant . However, there is a wind from the south with a constant speed of .
On what constant bearing, to the nearest degree, should the direction of the plane be set in order to reach the destination?
How to recognise this question
Question type: vectors
- The command and notation point to vectors.
- The requested response is a worked solution.
How to handle it: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Ground velocity must point on bearing . Wind is from the south, so wind velocity is due north at .
- Let the air-velocity heading make an angle east of the ground track (into the wind correction). In the velocity triangle, by the sine rule: .
- .
- Because the wind pushes the plane northward, the heading must be south of the ground track: bearing .
- NESA: 3 marks full; 2 for attempting the sine rule in the correct triangle; 1 for a suitable diagram.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Same destination bearing , airspeed , wind from the south at . Find the heading bearing to the nearest degree.
Question 2
Ground track bearing , airspeed , wind from the south . Find heading to nearest degree.
Question 3
Ground track , airspeed , wind from the north at . Explain how the correction angle changes.