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2023 HSC Mathematics Extension 1

Question 14(c):
Vectors

7 marksvectorsdependency-safe group

(i) Given a non-zero vector , the vector is perpendicular to it and has the same magnitude (Do NOT prove this).

Points and have position vectors and .

Using the given information, or otherwise, show that the area of triangle is .

(ii) Point lies on the circle centre radius with at angle to the horizontal. Point lies on the circle centre radius with at angle to the horizontal.

Note and .

Using part (i), find the values of that maximise the area of triangle .

Two circles tangent at O, with points P and Q defined by angles t and 2t at their respective centres.
Two circles tangent at O, with points P and Q defined by angles t and 2t at their respective centres.

How to recognise this question

Question type: vectors

  • The command and notation point to vectors.
  • The word “hence” means the later part is intended to reuse the earlier result.

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 restart the second part with unrelated numbers or discard the result proved in the first part.

Step-by-step answer

  1. (i) Let be the angle between and . Area .
  2. Take , perpendicular to with .
  3. Then (up to sign), and in absolute value, because .
  4. Hence area .
  5. NESA (i): 3 marks; 2 for relating the perpendicular vector's dot product to the area; 1 for attempting a perpendicular vector.
  6. (ii) , .
  7. Area .
  8. Factor : expand and use angle identities to obtain area (NESA: area on the interval where this expression is non-negative).
  9. Let . Then .
  10. Since , the sign of matches . Thus increases on and decreases on .
  11. By evenness of the area in , the area of is maximised at .
  12. NESA (ii): 4 marks; 3 for stationary values; 2 for area in terms of ; 1 for components of or .

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

The paper records 7 total marks 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

(i) Given that is perpendicular to with equal magnitude, show that the area of with and is .

(ii) Point lies on the circle centre radius with at angle to the horizontal. Point lies on the circle centre radius with also at angle to the horizontal. Using part (i), find that maximise the area of .

See the step-by-step answer guide →

Question 2

(i) Using a perpendicular vector of equal magnitude, show that the area of is .

(ii) Hence, with and , show the area of is (up to an equivalent non-negative form), and find maximising the area.

See the step-by-step answer guide →

Question 3

(i) Show that the area of with , is , and verify with , .

(ii) Hence, for arising as the area factor for the double-angle circle configuration of the exam, find all that maximise when (equivalently maximise the triangle area).

See the step-by-step answer guide →