Precalculus Vectors Quiz with Answers – Free PDF

Contents

This precalculus vectors quiz asks 10 multiple-choice questions on vectors in the plane, and each answer comes with a short explanation.

Ten multiple-choice questions on Precalculus vectors: magnitude, component form, direction angles, scalar multiples, unit vectors, dot products, and a river-crossing problem. Pick the one best answer for each question, then read the explanation.

PrecalculusCommon Core: N-VM.A.1 · N-VM.A.2 · N-VM.A.3 · N-VM.B.4 · N-VM.B.5

🎯 Vectors in the Plane – Precalculus math quiz

Score: 0 / 10

Answered: 0 / 10

Question 1

Find \(|\langle -5, 12\rangle|\).

A \(13\)

B \(7\)

C \(17\)

D \(169\)

Question 2

What is the component form of the vector from \(A(1, 2)\) to \(B(5, 5)\) in the figure?

Arrow from A to B.

A \(\langle 4, 3\rangle\)

B \(\langle 3, 4\rangle\)

C \(\langle 6, 7\rangle\)

D \(\langle -4, -3\rangle\)

Question 3

What is the direction angle of \(\langle -1, 1\rangle\)?

A \(45^{\circ}\)

B \(135^{\circ}\)

C \(225^{\circ}\)

D \(315^{\circ}\)

Question 4

Simplify \(3\langle 2, -1\rangle-\langle 1, 4\rangle\).

A \(\langle 5, 1\rangle\)

B \(\langle 7, -7\rangle\)

C \(\langle 5, 7\rangle\)

D \(\langle 5, -7\rangle\)

Question 5

Which is the unit vector in the direction of \(\langle 6, -8\rangle\)?

A \(\langle \frac{3}{5}, \frac{4}{5}\rangle\)

B \(\langle \frac{3}{5}, -\frac{4}{5}\rangle\)

C \(\langle -\frac{3}{5}, \frac{4}{5}\rangle\)

D \(\langle \frac{6}{5}, -\frac{8}{5}\rangle\)

Question 6

Find \(\langle 3, -2\rangle\cdot\langle 4, 5\rangle\).

A \(22\)

B \(2\)

C \(-2\)

D \(7\)

Question 7

What is the angle between the two vectors shown?

Two vectors from the origin.

A \(30^{\circ}\)

B \(60^{\circ}\)

C \(45^{\circ}\)

D \(90^{\circ}\)

Question 8

Which pair of vectors is perpendicular?

A \(\langle 2, 3\rangle\) and \(\langle 3, 2\rangle\)

B \(\langle 2, 3\rangle\) and \(\langle -2, 3\rangle\)

C \(\langle 2, 3\rangle\) and \(\langle 3, -2\rangle\)

D \(\langle 2, 3\rangle\) and \(\langle 2, -3\rangle\)

Question 9

The boat in the figure heads straight across at 12 mph and the current is 5 mph. What is its speed relative to the ground?

Boat heading across at 12 mph with a 5 mph current.

A \(7\text{ mph}\)

B \(17\text{ mph}\)

C \(60\text{ mph}\)

D \(13\text{ mph}\)

Question 10

If \(\vec{v}=\langle 3, 4\rangle\), what is \(|-3\vec{v}|\)?

A \(5\)

B \(21\)

C \(15\)

D \(-15\)

🎉 Quiz complete!

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How does this quiz work?

Read each question, choose one answer among A, B, C and D, then check it: a short explanation appears after every answer. Your score out of 10 is shown at the end, and you can restart the quiz as often as you like.

What should you review before the quiz?

Reread the key facts of the lesson and the worked example. If a question is hard, go back to the matching part of the lesson, then try the quiz again. Write your work on scrap paper: most mistakes come from skipping a step, not from the idea itself.

Remember:

Magnitude of ⟨a, b⟩ is √(a² + b²); direction angle satisfies tan θ = b/a in the correct quadrant.
Add and subtract component by component; the parallelogram rule gives the same sum geometrically.
Scalar multiplication k⟨a, b⟩ = ⟨ka, kb⟩ changes the length by |k| and reverses direction when k is negative.
A unit vector in the direction of v is v/|v|; every vector is a·i + b·j.
u·v = u₁v₁ + u₂v₂ = |u||v| cos θ, and u·v = 0 exactly when u and v are perpendicular.