2025 HSC Mathematics Extension 1
Question 14(c):
Vectors
At minutes after midnight, the clock-hand endpoints are and . Find the first two positive times when the hands are perpendicular, to three decimal places.
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
- Perpendicular hands have dot product zero.
- The dot product simplifies to .
- Thus .
- The first two positive times are and minutes.
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
Find the projection of onto .
Question 2
Find if is parallel to .
Question 3
Find the angle between and .