Worksheets · Precalculus

Vector basics worksheet

A vector is a size and a direction, and those are the two ways to write it down: components, or a magnitude with an angle. Everything on this sheet is moving between the two.

Select your difficulty

Pick more than one for a sheet that mixes them.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Vector Basics

Date Period

Find each answer. Round angles to the nearest tenth of a degree.

  1. Write PQ→ in component form: P(2,4), Q(3,13)
  2. Find the magnitude: ⟨−15,−8⟩
  3. Find the direction angle: ⟨−6,−8⟩
  4. Write PQ→ in component form: P(−8,−1), Q(−10,4)
  5. Find the magnitude: ⟨−12,−9⟩
  6. Find the direction angle: ⟨12,−9⟩
  7. Write PQ→ in component form: P(−6,−3), Q(−1,6)
  8. Find the magnitude: ⟨−8,15⟩
  9. Find the direction angle: ⟨−9,−12⟩
  10. Write PQ→ in component form: P(−3,3), Q(−8,12)
  11. Find the magnitude: ⟨16,12⟩
  12. Find the direction angle: ⟨−24,−10⟩
  13. Write PQ→ in component form: P(−6,3), Q(1,7)
  14. Find the magnitude: ⟨12,−16⟩
  15. Find the direction angle: ⟨−8,−15⟩
  16. Write PQ→ in component form: P(−4,−6), Q(−3,−14)
  17. Find the magnitude: ⟨−12,−5⟩
  18. Find the direction angle: ⟨−12,−9⟩
  19. Write PQ→ in component form: P(0,0), Q(−4,3)
  20. Find the magnitude: ⟨−8,6⟩

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Vector BasicsAnswer keyVersion 1

Date Period

Find each answer. Round angles to the nearest tenth of a degree.

  1. Write PQ→ in component form: P(2,4), Q(3,13)⟨1,9⟩
  2. Find the magnitude: ⟨−15,−8⟩17
  3. Find the direction angle: ⟨−6,−8⟩233.1∘
  4. Write PQ→ in component form: P(−8,−1), Q(−10,4)⟨−2,5⟩
  5. Find the magnitude: ⟨−12,−9⟩15
  6. Find the direction angle: ⟨12,−9⟩323.1∘
  7. Write PQ→ in component form: P(−6,−3), Q(−1,6)⟨5,9⟩
  8. Find the magnitude: ⟨−8,15⟩17
  9. Find the direction angle: ⟨−9,−12⟩233.1∘
  10. Write PQ→ in component form: P(−3,3), Q(−8,12)⟨−5,9⟩
  11. Find the magnitude: ⟨16,12⟩20
  12. Find the direction angle: ⟨−24,−10⟩202.6∘
  13. Write PQ→ in component form: P(−6,3), Q(1,7)⟨7,4⟩
  14. Find the magnitude: ⟨12,−16⟩20
  15. Find the direction angle: ⟨−8,−15⟩241.9∘
  16. Write PQ→ in component form: P(−4,−6), Q(−3,−14)⟨1,−8⟩
  17. Find the magnitude: ⟨−12,−5⟩13
  18. Find the direction angle: ⟨−12,−9⟩216.9∘
  19. Write PQ→ in component form: P(0,0), Q(−4,3)⟨−4,3⟩
  20. Find the magnitude: ⟨−8,6⟩10

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the magnitude and direction of ⟨−5,12⟩

  1. The magnitude is the square root of 25 plus 144.
  2. That is the square root of 169, which is 13.
  3. The angle is arctan of 12 over -5, in the second quadrant.

The answer is 13, 112.6∘.

Where students go wrong

Reading the direction angle straight off the calculator. Arctangent only ever answers between -90 and 90 degrees, so for a vector pointing left the calculator is 180 degrees out. Look at which quadrant the components put you in first.

Also called component form, magnitude and direction, i and j form or resultant vectors.

What you can put on this worksheet

Component form from two points
Write PQ→ in component form: P(−3,5), Q(2,11) → ⟨5,6⟩
The magnitude of a vector
Find the magnitude: ⟨12,−5⟩ → 13
The direction angle
Find the direction angle: ⟨−6,0⟩ → 180.0∘
Component form from a magnitude and an angle
Write in component form: ∣v⃗∣=8, θ=120∘ → ⟨−4,43⟩
Between component form and i, j form
Write in component form: 12i⃗−5j⃗ → ⟨12,−5⟩

Questions about these worksheets

Yes. Every vector basics sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the vector basics questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: component form from two points, the magnitude of a vector, the direction angle, component form from a magnitude and an angle and between component form and i, j form. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the direction angle straight off the calculator. Arctangent only ever answers between -90 and 90 degrees, so for a vector pointing left the calculator is 180 degrees out. Look at which quadrant the components put you in first.