Vectors — Difference of vectors and scalar product

Question

Let and .

(i) Find .

(ii) Find .

This practice question was inspired by Question 11(a) 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.(i) , so .
Finish the calculation, then check that the result meets the question’s conditions.(ii) .
State the final answer clearly in the form the question requested.Answer: (i) ; (ii) .

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

Verification: Component arithmetic: and .

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