Vectors — Dot identity with vector components stated

Question

(i) For a vector , show that .

(ii) In the trapezium , and . Let , and with . Using part (i), show that .

This practice question was inspired by Question 14(c) in the 2021 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) .
Simplify the previous line carefully, keeping exact values where possible.(ii) , .
Simplify the previous line carefully, keeping exact values where possible.Using (i): .
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested.Divide by : .

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

Verification: (i) Component identity in 3D. (ii) Same diagonal-equality algebra as the exam.

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