Worksheets · Precalculus

Vector operations worksheet

Vectors add component by component, and a scalar multiplies every component. A unit vector is the same direction shrunk to length one, which is what dividing by the magnitude does.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Vector Operations

Date Period

Find each answer. Round decimal answers and angles to the nearest tenth.

  1. Add, subtract and scale vectors

    Find 4u+3v: u=⟨3,−10⟩, v=⟨1,10⟩
  2. Find 3u−3v: u=⟨1,5⟩, v=⟨−6,5⟩
  3. Find 3u+3v: u=⟨−8,−2⟩, v=⟨−4,3⟩
  4. Find 4u−4v: u=⟨−4,−6⟩, v=⟨−5,9⟩

  1. Find the unit vector

    Find the unit vector in the direction of v=⟨−12,−16⟩
  2. Find the unit vector in the direction of v=⟨8,−6⟩
  3. Find the unit vector in the direction of v=⟨−24,7⟩
  4. Find the unit vector in the direction of v=⟨−15,−8⟩

  1. Find the opposite unit vector

    Find the unit vector opposite v=⟨−8,−15⟩
  2. Find the unit vector opposite v=⟨5,12⟩
  3. Find the unit vector opposite v=⟨12,−5⟩
  4. Find the unit vector opposite v=⟨−20,21⟩

  1. Find a vector of a given length

    Find the vector with magnitude 10 in the direction of u=⟨−3,4⟩
  2. Find the vector with magnitude 100 in the direction opposite to u=⟨24,7⟩
  3. Find the vector with magnitude 26 in the direction of u=⟨12,5⟩
  4. Find the vector with magnitude 52 in the direction opposite to u=⟨12,−5⟩

  1. Find the magnitude and direction of u+v\mathbf{u} + \mathbf{v}

    u=⟨−6,3⟩, v=⟨11,9⟩. Find the magnitude and direction angle of u+v
  2. u=⟨7,0⟩, v=⟨−10,4⟩. Find the magnitude and direction angle of u+v
  3. u=⟨−5,−2⟩, v=⟨20,10⟩. Find the magnitude and direction angle of u+v
  4. u=⟨8,−3⟩, v=⟨−4,6⟩. Find the magnitude and direction angle of u+v

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

Sigma Prep · sigmaprep.io/worksheets

Name

Vector OperationsAnswer keyVersion 1

Date Period

Find each answer. Round decimal answers and angles to the nearest tenth.

  1. Add, subtract and scale vectors

    Find 4u+3v: u=⟨3,−10⟩, v=⟨1,10⟩⟨15,−10⟩
  2. Find 3u−3v: u=⟨1,5⟩, v=⟨−6,5⟩⟨21,0⟩
  3. Find 3u+3v: u=⟨−8,−2⟩, v=⟨−4,3⟩⟨−36,3⟩
  4. Find 4u−4v: u=⟨−4,−6⟩, v=⟨−5,9⟩⟨4,−60⟩

  1. Find the unit vector

    Find the unit vector in the direction of v=⟨−12,−16⟩⟨−35,−45⟩
  2. Find the unit vector in the direction of v=⟨8,−6⟩⟨45,−35⟩
  3. Find the unit vector in the direction of v=⟨−24,7⟩⟨−2425,725⟩
  4. Find the unit vector in the direction of v=⟨−15,−8⟩⟨−1517,−817⟩

  1. Find the opposite unit vector

    Find the unit vector opposite v=⟨−8,−15⟩⟨817,1517⟩
  2. Find the unit vector opposite v=⟨5,12⟩⟨−513,−1213⟩
  3. Find the unit vector opposite v=⟨12,−5⟩⟨−1213,513⟩
  4. Find the unit vector opposite v=⟨−20,21⟩⟨2029,−2129⟩

  1. Find a vector of a given length

    Find the vector with magnitude 10 in the direction of u=⟨−3,4⟩⟨−6,8⟩
  2. Find the vector with magnitude 100 in the direction opposite to u=⟨24,7⟩⟨−96,−28⟩
  3. Find the vector with magnitude 26 in the direction of u=⟨12,5⟩⟨24,10⟩
  4. Find the vector with magnitude 52 in the direction opposite to u=⟨12,−5⟩⟨−48,20⟩

  1. Find the magnitude and direction of u+v\mathbf{u} + \mathbf{v}

    u=⟨−6,3⟩, v=⟨11,9⟩. Find the magnitude and direction angle of u+vmagnitude 13, direction 67.4∘
  2. u=⟨7,0⟩, v=⟨−10,4⟩. Find the magnitude and direction angle of u+vmagnitude 5, direction 126.9∘
  3. u=⟨−5,−2⟩, v=⟨20,10⟩. Find the magnitude and direction angle of u+vmagnitude 17, direction 28.1∘
  4. u=⟨8,−3⟩, v=⟨−4,6⟩. Find the magnitude and direction angle of u+vmagnitude 5, direction 36.9∘

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

How to do these

2u−3v where u=⟨4,−1⟩, v=⟨2,3⟩

  1. Two times u is <8, -2>.
  2. Three times v is <6, 9>.
  3. Subtract component by component.

The answer is ⟨2,−11⟩.

Where students go wrong

Subtracting the magnitudes instead of the components. The length of u minus v has almost nothing to do with the length of u minus the length of v; you have to do the subtraction first and take the magnitude of what comes out.

Also called adding vectors, scalar multiplication, unit vectors or resultant vector.

What you can put on this worksheet

Sums, differences and scalar multiples
Find u+v: u=⟨1,1⟩, v=⟨4,−7⟩ → ⟨5,−6⟩
The unit vector in the same direction
Find the unit vector in the direction of v=⟨12,−5⟩ → ⟨1213,−513⟩
The unit vector in the opposite direction
Find the unit vector opposite v=⟨12,−5⟩ → ⟨−1213,513⟩
The magnitude of a sum
Find ∣u+v∣: u=⟨0,1⟩, v=⟨12,−6⟩ → 13
The opposite vector
Find the opposite of v=⟨6,−1⟩ → −v=⟨−6,1⟩
A given length along or against a vector
Find the vector with magnitude 51 in the direction opposite to u=⟨8,15⟩ → ⟨−24,−45⟩
Magnitude and direction of a sum
u=⟨0,1⟩, v=⟨5,11⟩. Find the magnitude and direction angle of u+v → magnitude 13, direction 67.4∘
Combinations of vectors between points
Find −3AB→+3CD→ for these points:
A(0,1), B(2,−4), C(0,1), D(−2,4) → ⟨−12,24⟩
Unit and opposite vectors from points
Find the unit vector in the direction of AB→: A(1,3),B(4,−1) → ⟨35,−45⟩
Magnitude and direction of a combination
u=⟨0,1⟩, v=⟨3,−5⟩. Find the magnitude and direction angle of −2u−3v → ⟨−9,13⟩; magnitude 510≈15.8, direction 124.7∘

Questions about these worksheets

Yes. Every vector operations 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 operations 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 10: sums, differences and scalar multiples, the unit vector in the same direction, the unit vector in the opposite direction, the magnitude of a sum, the opposite vector, a given length along or against a vector, magnitude and direction of a sum, combinations of vectors between points, unit and opposite vectors from points and magnitude and direction of a combination. Tick as many as you want and set how many of each, or let it spread them evenly.

Subtracting the magnitudes instead of the components. The length of u minus v has almost nothing to do with the length of u minus the length of v; you have to do the subtraction first and take the magnitude of what comes out.