Vectors — Dot identity and parallelogram case

Question

(i) For a vector , show that .

(ii) In the trapezium , is parallel to and . Let , and , where . Using part (i), or otherwise, show that . Hence deduce that if , then .

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) If , then . Alternatively .
Simplify the previous line carefully, keeping exact values where possible.(ii) Diagonals: and .
Simplify the previous line carefully, keeping exact values where possible.Equal lengths equal squared lengths (using (i)): .
Simplify the previous line carefully, keeping exact values where possible.Expand: .
Simplify the previous line carefully, keeping exact values where possible.Simplify: .
Finish the calculation, then check that the result meets the question’s conditions.Since , : , so .
State the final answer clearly in the form the question requested.If , this reduces to (a rectangle).

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

Verification: (i) . (ii) Squared diagonals yield the stated relation; forces .

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