Vectors — Similar quadratic

Question

Find so that and are perpendicular.

This practice question was inspired by Question 11(d) 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..
State the final answer clearly in the form the question requested.Discriminant : no real .

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

Verification: Dot product zero may have 0, 1, or 2 real roots.

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