Vectors — Find x from two projections

Question

Let and . If and , find in terms of and .

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.Equations: and .
Finish the calculation, then check that the result meets the question’s conditions.Solve: , .
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: Clearing denominators produces the NESA linear system.

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