2024 HSC Mathematics Extension 1
Question 13(c):
Vectors
The vector is and the vector is .
The projection of a vector onto the vector is , where is a real number.
The projection of the vector onto the vector is , where is a real number.
Find the vector in terms of and .
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
- Let .
- Projection formula: , so i.e. .
- Similarly i.e. .
- Solve the linear system: from the second, . Substitute: .
- Then .
- Hence .
- NESA 13(c): 4 marks correct; 3 marks for both and in terms of components; 2 marks for one of them; 1 mark for writing a projection formula.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 4 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
Let and . If and , find in terms of and .
Question 2
If and , write the scalar such that , and similarly for . Then solve for in terms of .
Question 3
Let and . If and , find in terms of and .