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2021 HSC Mathematics Extension 1

Question 5:
Vectors

1 markvectors

For the two vectors and it is known that .

Which of the following statements MUST be true?

  1. Either is negative and is positive, or is positive and is negative.
  2. The angle between and is obtuse.
  3. The product is negative.
  4. The points , and are collinear.

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. By definition . For nonzero vectors the magnitudes are positive, so the sign of the dot product equals the sign of .
  2. If the dot product is negative then , so the angle between the vectors is obtuse.
  3. Option A confuses components with an undefined notion of a "positive vector". Option C is impossible because magnitudes are non-negative. Option D is not forced by a negative dot product.
  4. 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

If for nonzero vectors , which statement MUST be true?

  1. The angle between and is acute
  2. The angle between and is obtuse
  3. The angle is a right angle
  4. is negative
See the step-by-step answer guide →

Question 2

If and are nonzero, which MUST be true?

  1. The vectors are perpendicular
  2. The vectors are parallel
  3. The angle is obtuse
  4. One vector must be the zero vector
See the step-by-step answer guide →

Question 3

For and , which is true?

  1. so the angle is obtuse
  2. so the angle is acute
  3. so the angle is right
  4. The angle cannot be determined from the components
See the step-by-step answer guide →