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MCV4U Cartesian Vectors, Dot & Cross Products
Grade 12 Unit 6 — practice questions with full solutions

Vectors in component form: dot products and angles, cross products, areas, and projections in three dimensions.

Free MCV4U cartesian vectors, dot & cross products practice for Grade 12 students in Ontario. These 6 questions cover the same material as a typical unit test or exam review on this topic, and every one comes with a complete worked solution — not just an answer key. Useful whether you are preparing for a MCV4U unit test, catching up on a lesson, or reviewing before the final exam.
MCV4U · Grade 12 Geometry and Algebra of Vectors 5 core 1 challenge
1Core
Given a = (2, −1, 3) and b = (4, 0, −2), find the dot product a · b.
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  1. The dot product multiplies matching components and adds the results.
  2. (2)(4) = 8.
  3. (−1)(0) = 0.
  4. (3)(−2) = −6.
  5. Sum: 8 + 0 − 6 = 2.
Final answer2
2Core
Using a = (2, −1, 3) and b = (4, 0, −2), find the angle between the two vectors, to one decimal place.
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  1. Use cos θ = (a · b) ÷ (|a||b|), with a · b = 2.
  2. |a| = √(4 + 1 + 9) = √14 ≈ 3.7417.
  3. |b| = √(16 + 0 + 4) = √20 ≈ 4.4721.
  4. cos θ = 2 ÷ (3.7417 × 4.4721) ≈ 0.1195.
  5. θ = arccos(0.1195) ≈ 83.1°.
Final answer≈ 83.1°
3Core
Given a = (2, −1, 3) and b = (4, 0, −2), find the cross product a × b, then verify your answer is perpendicular to a.
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  1. First component: (−1)(−2) − (3)(0) = 2 − 0 = 2.
  2. Second component: −[(2)(−2) − (3)(4)] = −[−4 − 12] = 16.
  3. Third component: (2)(0) − (−1)(4) = 0 + 4 = 4.
  4. So a × b = (2, 16, 4).
  5. Check perpendicularity: (2)(2) + (−1)(16) + (3)(4) = 4 − 16 + 12 = 0. ✔
Final answera × b = (2, 16, 4)
4Core
Find the area of the parallelogram determined by a = (2, −1, 3) and b = (4, 0, −2), to two decimal places.
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  1. The area of the parallelogram equals the magnitude of the cross product.
  2. From the previous result, a × b = (2, 16, 4).
  3. |a × b| = √(4 + 256 + 16) = √276.
  4. √276 ≈ 16.61 square units.
Final answer√276 ≈ 16.61 square units
5Core
Find the projection of a = (2, −1, 3) onto b = (4, 0, −2).
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  1. The projection is [(a · b) ÷ |b|2]b.
  2. a · b = 2 and |b|2 = 16 + 0 + 4 = 20.
  3. The scalar factor is 2 ÷ 20 = 0.1.
  4. Multiply b by 0.1: (0.4, 0, −0.2).
Final answer(0.4, 0, −0.2)
6Challenge
Find the value of k that makes (k, 2, −1) perpendicular to (3, k, 4).
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  1. Two vectors are perpendicular exactly when their dot product is zero.
  2. (k)(3) + (2)(k) + (−1)(4) = 0.
  3. 3k + 2k − 4 = 0, so 5k = 4.
  4. k = 4⁄5 = 0.8.
Final answerk = 4⁄5

MCV4U Cartesian Vectors, Dot & Cross Products — common questions

Short answers to the things students ask most about this unit.

What does a dot product of zero mean?

It means the two vectors are perpendicular. This is the quickest test for perpendicularity and appears often on unit tests.

What is the cross product used for?

The cross product produces a vector perpendicular to both originals, and its magnitude equals the area of the parallelogram they determine.

More MCV4U practice — grade 12 calculus and vectors

MCV4U Rates of Change & LimitsMCV4U DerivativesMCV4U Applications of Derivatives & Curve SketchingMCV4U Derivatives of Exponential & Trigonometric FunctionsAll MCV4U units
← Unit 5: Geometric VectorsUnit 7: Equations of Lines & Planes →
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