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2022 HSC Mathematics Extension 1

Question 6:
Vectors

1 markvectors

The following diagram shows the vector and the vectors , , and . (Drawn to scale: lies in the first quadrant, steeper than .)

Which statement regarding this diagram could be true?

The drawn-to-scale vector diagram supplied with the original question.
The drawn-to-scale vector diagram supplied with the original question.
  1. The projection of onto is the vector .
  2. The projection of onto is the vector .
  3. The projection of onto is the vector .
  4. The projection of onto is the vector .

How to recognise this question

Question type: vectors

  • The command and notation point to vectors.
  • The requested response is a multiple choice.

How to handle it: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. A projection of onto a direction must be a scalar multiple of (same or opposite direction).
  2. A: is not parallel to (components unequal).
  3. C: is parallel to , not to ; also far too long relative to the scale of .
  4. D: is parallel to , but in the first quadrant has negative (or near-zero) projection onto the fourth-quadrant diagonal , not a substantial positive one as drawn.
  5. B: is parallel to and a short vector in that direction, consistent with a small positive projection of onto the second-quadrant diagonal.
  6. Answer: B. (NESA multiple-choice key: B.)

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.

Try your hand at it again by answering these questions

Question 1

Which vector could be the projection of some onto ?

See the step-by-step answer guide →

Question 2

The vector cannot be the projection of onto

  2. (same line, but check orientation/length separately)
  4. a positive multiple of
See the step-by-step answer guide →

Question 3

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
See the step-by-step answer guide →