27 exam-style questions on Scalar and vector product; angles; interpretation — topic 3.13 of Geometry and Trigonometry in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
Calculate the scalar (dot) product of a = (3, 4) and b = (2, −1).
Answer: 2
a · b = (3)(2) + (4)(−1) = 6 − 4 = 2. The dot product is a scalar, not a vector like (6, −4).
Determine the angle between a = (1, 0) and b = (1, 1), to the nearest degree.
Answer: 45°
cos θ = (a · b)/(|a||b|) = 1/(1 × √2) = 1/√2, so θ = 45°.
Given |a| = 4, |b| = 3 and the angle between them is 60°, calculate a · b.
Answer: 6
a · b = |a||b|cos θ = 4 × 3 × cos 60° = 12 × 0.5 = 6. Leaving out cos θ gives 12; using sin 60° instead gives 10.4.
Calculate the vector product a × b for a = (1, 0, 0) and b = (0, 1, 0).
Answer: (0, 0, 1)
The vector product of two perpendicular unit vectors i × j = k = (0, 0, 1), a vector perpendicular to both.
Two adjacent sides of a parallelogram are given by a = (3, 0, 0) and b = (0, 4, 0). Determine the area of the parallelogram.
Answer: 12
Area = |a × b| = |a||b|sin θ = 3 × 4 × sin 90° = 12. The vectors are perpendicular so the parallelogram is a rectangle of area 12.
Given a · b = 8, |a| = 4 and |b| = 4, determine the angle between a and b, to the nearest degree.
Answer: 60°
cos θ = (a · b)/(|a||b|) = 8/(4 × 4) = 8/16 = 0.5, so θ = 60°.
That was 6 of 27 Scalar and vector product; angles; interpretation questions.
Practise all 27 freeCalculateAO2 · 2–6 marks
Show the formula, the substitution and the answer with units and sensible significant figures.
DetermineAO2 · 2–6 marks
Obtain the only possible answer — by calculation, from the graph or by reasoning — and show how.
StateAO1 · 1–2 marks
One word or one short sentence. No explanation, no working: the mark is for the fact itself.
27 Scalar and vector product; angles; interpretation questions — free with an account. Spaced repetition, streaks and full exam simulations included.
Practise them freeThis platform is independently developed and is not endorsed by, affiliated with, or sponsored by the International Baccalaureate Organization. “International Baccalaureate” and “IB” are registered trademarks of the IBO.