2022 HSC Mathematics Extension 1
Question 13(a):
Vectors
Three different points , and are chosen on a circle centred at .
Let , and . Let and let be the point such that .
Show that and are perpendicular.
How to recognise this question
Question type: vectors
- The command and notation point to vectors.
- The requested response is a show that.
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
- and .
- Dot product: .
- But (both radii), so the difference of squares is 0.
- Hence , so .
- NESA: 3 marks; 2 for attempting the dot product in terms of ; 1 for expressing or in vector form.
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
With the same setup, show .
Question 2
Explain why and are perpendicular when .
Question 3
If , (equal length 5), verify .