2022 HSC Mathematics Extension 1
Question 14(b):
Vectors
The vectors and are not parallel. The vector is the projection of onto .
The vector is parallel to , so for some real . (Do NOT prove this.)
Prove that is smallest when by showing that for all real , .
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
- Write .
- The remainder is orthogonal to (property of projection), so it is orthogonal to .
- Expand the squared norm: (cross terms vanish).
- The second term is and equals 0 iff (since ).
- Therefore , hence for all real .
- NESA: 3 marks; 2 for a suitable inequality via a vector method; 1 for writing as parallel plus perpendicular parts.
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
Explain the same result using a right triangle formed by , its projection , and on the line of .
Question 2
For fixed , let . Show is minimised at .
Question 3
When does equality hold in ?