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.

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Pick more than one for a sheet that mixes them.

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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 a vector from two points

    Write PQ→ in component form: P(2,4), Q(3,13)
  2. Write PQ→ in component form: P(5,−3), Q(−1,−12)
  3. Write PQ→ in component form: P(4,−2), Q(6,5)
  4. Write PQ→ in component form: P(0,−5), Q(3,−2)
  5. Write PQ→ in component form: P(−5,−3), Q(−12,−12)

  1. Find the magnitude of a vector

    Find the magnitude: ⟨−15,−8⟩
  2. Find the magnitude: ⟨8,6⟩
  3. Find the magnitude: ⟨−5,12⟩
  4. Find the magnitude: ⟨24,−7⟩
  5. Find the magnitude: ⟨−20,−21⟩

  1. Find the direction angle

    Find the direction angle: ⟨−6,−8⟩
  2. Find the direction angle: ⟨−8,6⟩
  3. Find the direction angle: ⟨5,12⟩
  4. Find the direction angle: ⟨8,−6⟩
  5. Find the direction angle: ⟨8,−15⟩

  1. Find horizontal and vertical components

    A sled is pulled with a force of 15 pounds along a rope at 38∘ to the ground. Find the horizontal and vertical components, to the nearest tenth.
  2. A plane climbs at 199 mph at 62∘ above the horizontal. Find the horizontal and vertical components, to the nearest tenth.
  3. A ball is thrown at 19 ft/s at 21∘ above the horizontal. Find the horizontal and vertical components, to the nearest tenth.
  4. A sled is pulled with a force of 39 pounds along a rope at 35∘ to the ground. Find the horizontal and vertical components, to the nearest tenth.
  5. A plane climbs at 284 mph at 34∘ above the horizontal. Find the horizontal and vertical components, to the nearest tenth.

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 a vector from two points

    Write PQ→ in component form: P(2,4), Q(3,13)⟨1,9⟩
  2. Write PQ→ in component form: P(5,−3), Q(−1,−12)⟨−6,−9⟩
  3. Write PQ→ in component form: P(4,−2), Q(6,5)⟨2,7⟩
  4. Write PQ→ in component form: P(0,−5), Q(3,−2)⟨3,3⟩
  5. Write PQ→ in component form: P(−5,−3), Q(−12,−12)⟨−7,−9⟩

  1. Find the magnitude of a vector

    Find the magnitude: ⟨−15,−8⟩17
  2. Find the magnitude: ⟨8,6⟩10
  3. Find the magnitude: ⟨−5,12⟩13
  4. Find the magnitude: ⟨24,−7⟩25
  5. Find the magnitude: ⟨−20,−21⟩29

  1. Find the direction angle

    Find the direction angle: ⟨−6,−8⟩233.1∘
  2. Find the direction angle: ⟨−8,6⟩143.1∘
  3. Find the direction angle: ⟨5,12⟩67.4∘
  4. Find the direction angle: ⟨8,−6⟩323.1∘
  5. Find the direction angle: ⟨8,−15⟩298.1∘

  1. Find horizontal and vertical components

    A sled is pulled with a force of 15 pounds along a rope at 38∘ to the ground. Find the horizontal and vertical components, to the nearest tenth.horizontal 11.8 pounds, vertical 9.2 pounds
  2. A plane climbs at 199 mph at 62∘ above the horizontal. Find the horizontal and vertical components, to the nearest tenth.horizontal 93.4 mph, vertical 175.7 mph
  3. A ball is thrown at 19 ft/s at 21∘ above the horizontal. Find the horizontal and vertical components, to the nearest tenth.horizontal 17.7 ft/s, vertical 6.8 ft/s
  4. A sled is pulled with a force of 39 pounds along a rope at 35∘ to the ground. Find the horizontal and vertical components, to the nearest tenth.horizontal 31.9 pounds, vertical 22.4 pounds
  5. A plane climbs at 284 mph at 34∘ above the horizontal. Find the horizontal and vertical components, to the nearest tenth.horizontal 235.4 mph, vertical 158.8 mph

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⟩
Graph a vector on a grid
Graph ⟨4,1⟩ from the origin. → Ending at (4,1)
Components in a word problem
A sled is pulled with a force of 55 pounds along a rope at 61∘ to the ground. Find the horizontal and vertical components, to the nearest tenth. → horizontal 26.7 pounds, vertical 48.1 pounds
Component form from a drawing
Write the vector shown in component form. → ⟨3,−8⟩
From a magnitude and any direction angle
A rope pulls on a crate with a force of 34 lb at a direction angle of 25∘. Find the horizontal and vertical components of the pull, to the nearest tenth. → ⟨30.8,14.4⟩

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 9: component form from two points, the magnitude of a vector, the direction angle, component form from a magnitude and an angle, between component form and i, j form, graph a vector on a grid, components in a word problem, component form from a drawing and from a magnitude and any direction angle. 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.