Vectors — Projection scalar setup

Question

If and , write the scalar such that , and similarly for . Then solve for in terms of .

This practice question was inspired by Question 13(c) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required., .
Finish the calculation, then check that the result meets the question’s conditions.System , yields the same as above.
State the final answer clearly in the form the question requested.Answer: .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Denominator for is .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.