MCV4U Vectors Practice: Components, Magnitude and Dot Product

To solve a vector problem, identify whether the answer should be a vector or a scalar, choose a consistent coordinate system, and write the relevant relationship before calculating. Component addition produces a vector; magnitude and dot product produce scalars.

This original MCV4U resource practises selected vector foundations. It is not a complete Calculus and Vectors test: line and plane equations, projections and other course topics need additional practice according to your teacher’s outline.

Why do vectors feel different from earlier algebra?

Vectors combine numerical work with direction and geometry. A student discussion describes knowing formulas but struggling when assessment questions look unfamiliar. That experience suggests a useful practice task: decide what the answer represents before selecting a formula. It does not establish that vectors are hardest for every student. Student discussion

Write a sentence such as “I need a direction vector from A to B” or “I need the angle between two nonzero vectors.” That sentence determines which operation is relevant.

Core relationships

• From A(x₁, y₁) to B(x₂, y₂), vector AB = (x₂ − x₁, y₂ − y₁).
• The magnitude of (a, b) is √(a² + b²); in three dimensions, include the third squared component.
• For u = (a, b) and v = (c, d), u · v = ac + bd.
• For nonzero vectors, u · v = |u||v|cos θ.

The zero vector has magnitude zero and no defined direction. The angle formula requires nonzero vectors because it divides by both magnitudes.

Eight original questions with solutions

1. Find a displacement vector

Let A = (1, −2) and B = (5, 1). Find AB and its magnitude.

Subtract initial coordinates from final coordinates: AB = (5 − 1, 1 − (−2)) = (4, 3). Its magnitude is √(16 + 9) = 5. BA would point the other way and equal (−4, −3).

2. Combine vectors component by component

For u = (2, −1) and v = (−3, 4), find 2u + v.

2u + v = (4, −2) + (−3, 4) = (1, 2). The result remains a vector. Multiplication by 2 scales both components.

3. Find a unit vector

Find a unit vector in the direction of w = (−6, 8).

4. Test perpendicularity

Are u = (3, 2) and v = (4, −6) perpendicular?

u · v = 3(4) + 2(−6) = 12 − 12 = 0. Both vectors are nonzero, so they are perpendicular. The dot product is a scalar, not an ordered pair.

5. Find an angle

Find the angle between u = (1, 0) and v = (1, √3).

The dot product is 1 and the magnitudes are 1 and 2. Thus cos θ = 1/(1 × 2) = 1/2, giving θ = 60°. State the angle unit; do not mix degree and radian calculator settings.

6. Use a perpendicularity condition to find a parameter

Find k if (k, 2) is perpendicular to (4, −6).

Set the dot product to zero: 4k − 12 = 0, so k = 3. Check by substitution: (3, 2) · (4, −6) = 0.

7. Interpret opposite direction in three dimensions

Given a = (1, 2, 2) and b = (−2, −4, −4), find their relationship and magnitudes.

b = −2a, so they are parallel and point in opposite directions. |a| = √(1 + 4 + 4) = 3 and |b| = 6. The angle between them is 180°.

8. Apply the dot product to work

A constant force F = (10, 0) N acts over displacement d = (3, 4) m. Find the work done by this force.

W = F · d = 10(3) + 0(4) = 30 J. The displacement magnitude is 5 m, but multiplying 10 N by 5 m would ignore the angle between the force and displacement. Only the component along the force contributes to this dot product.

How to practise for unfamiliar questions

Change one feature of a solved problem: reverse the vector direction, move a point into another quadrant, or ask for a parameter instead of an angle. Before calculating, predict what should change. Reversing one vector changes the sign of a dot product but not its magnitude.

Check dimensions and answer type. A direction vector should not be reported as a single length. An angle calculation should not involve dividing by zero. A purported unit vector should have magnitude 1.

Continue your senior math preparation

Read the MCV4U guide, explore Grade 12 math tutoring, or book a free one-hour assessment.

Curriculum reference: Ontario Grades 11–12 Mathematics.