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← 2022 HSC Extension 1 paper Question 6

Variation 3 · answer guide

Vectors — Signed projection length

Question

If lies in the first quadrant and , the projection of onto

  1. points into the third quadrant (negative multiple of sense along )
  2. must be
  3. must be the zero vector
  4. must equal

This practice question was inspired by Question 6 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.

Step by step

  1. Set up the solution: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required.Dot product for first-quadrant and third-quadrant .
  2. Finish the calculation, then check that the result meets the question’s conditions.Projection is a negative multiple of 's unit direction, still along the third-quadrant line.
  3. Match the result to the choices and state the correct option.Therefore option A is correct.

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

Verification: Sign of fixes the projection's direction along the line.

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

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