Vectors — Negative components

Question

Let and . Evaluate each of the following.

(i)

(ii)

This practice question was inspired by Question 11(a) in the 2022 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) .
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: Watch signs on both the linear combination and the product.

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