Worksheets · Precalculus

Dot product worksheet

The dot product multiplies matching components and adds them, so it turns two vectors into one number. That number carries the angle between them, which is why the same formula answers both.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Dot Products

Date Period

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

  1. Find the dot product

    Find u⃗⋅v⃗: u⃗=⟨2,−7⟩, v⃗=⟨0,7⟩
  2. Find u⃗⋅v⃗: u⃗=⟨−2,4⟩, v⃗=⟨−3,−5⟩
  3. Find u⃗⋅v⃗: u⃗=⟨−1,−4⟩, v⃗=⟨−5,3⟩
  4. Find u⃗⋅v⃗: u⃗=⟨6,0⟩, v⃗=⟨7,3⟩
  5. Find u⃗⋅v⃗: u⃗=⟨−3,3⟩, v⃗=⟨5,−6⟩

  1. Find the angle between two vectors

    Find the angle between u⃗ and v⃗: u⃗=⟨14,−6⟩, v⃗=⟨3,6⟩
  2. Find the angle between u⃗ and v⃗: u⃗=⟨−13,−2⟩, v⃗=⟨3,9⟩
  3. Find the angle between u⃗ and v⃗: u⃗=⟨2,−11⟩, v⃗=⟨−7,−13⟩
  4. Find the angle between u⃗ and v⃗: u⃗=⟨−7,11⟩, v⃗=⟨9,8⟩

  1. Classify vectors: parallel, orthogonal or neither

    u⃗=⟨4,−2⟩, v⃗=⟨6,12⟩. Parallel, orthogonal, or neither?
  2. u⃗=⟨2,6⟩, v⃗=⟨−18,6⟩. Parallel, orthogonal, or neither?
  3. u⃗=⟨3,−6⟩, v⃗=⟨9,−18⟩. Parallel, orthogonal, or neither?
  4. u⃗=⟨−2,−5⟩, v⃗=⟨−4,−10⟩. Parallel, orthogonal, or neither?

  1. Find the work done by a force

    A force of 30 lb at 30∘ to the horizontal pulls a wagon 40 ft. Find the work done, to the nearest tenth.
  2. A rope at 45∘ to the water tows a kayak 100 ft with a pull of 75 lb. Find the work done, to the nearest tenth.
  3. A worker pushes a lawn mower 20 ft with 20 lb along the handle, which is at 45∘ below the horizontal. Find the work done, to the nearest tenth.
  4. A shopper pushes a cart 110 ft with 55 lb along a handle at 20∘ below the horizontal. Find the work done, to the nearest tenth.
  5. A child drags a sled 160 ft with a rope at 30∘ to the ground, pulling with 10 lb. Find the work done, to the nearest tenth.

  1. Decide whether vectors are orthogonal

    Are u⃗ and v⃗ orthogonal? u⃗=⟨−2,−3⟩, v⃗=⟨−3,−5⟩
  2. Are u⃗ and v⃗ orthogonal? u⃗=⟨−9,5⟩, v⃗=⟨−5,−12⟩

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

Sigma Prep · sigmaprep.io/worksheets

Name

Dot ProductsAnswer keyVersion 1

Date Period

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

  1. Find the dot product

    Find u⃗⋅v⃗: u⃗=⟨2,−7⟩, v⃗=⟨0,7⟩−49
  2. Find u⃗⋅v⃗: u⃗=⟨−2,4⟩, v⃗=⟨−3,−5⟩−14
  3. Find u⃗⋅v⃗: u⃗=⟨−1,−4⟩, v⃗=⟨−5,3⟩−7
  4. Find u⃗⋅v⃗: u⃗=⟨6,0⟩, v⃗=⟨7,3⟩42
  5. Find u⃗⋅v⃗: u⃗=⟨−3,3⟩, v⃗=⟨5,−6⟩−33

  1. Find the angle between two vectors

    Find the angle between u⃗ and v⃗: u⃗=⟨14,−6⟩, v⃗=⟨3,6⟩86.6∘
  2. Find the angle between u⃗ and v⃗: u⃗=⟨−13,−2⟩, v⃗=⟨3,9⟩117.2∘
  3. Find the angle between u⃗ and v⃗: u⃗=⟨2,−11⟩, v⃗=⟨−7,−13⟩38.6∘
  4. Find the angle between u⃗ and v⃗: u⃗=⟨−7,11⟩, v⃗=⟨9,8⟩80.8∘

  1. Classify vectors: parallel, orthogonal or neither

    u⃗=⟨4,−2⟩, v⃗=⟨6,12⟩. Parallel, orthogonal, or neither?Orthogonal
  2. u⃗=⟨2,6⟩, v⃗=⟨−18,6⟩. Parallel, orthogonal, or neither?Orthogonal
  3. u⃗=⟨3,−6⟩, v⃗=⟨9,−18⟩. Parallel, orthogonal, or neither?Parallel
  4. u⃗=⟨−2,−5⟩, v⃗=⟨−4,−10⟩. Parallel, orthogonal, or neither?Parallel

  1. Find the work done by a force

    A force of 30 lb at 30∘ to the horizontal pulls a wagon 40 ft. Find the work done, to the nearest tenth.1039.2 ft⋅lb
  2. A rope at 45∘ to the water tows a kayak 100 ft with a pull of 75 lb. Find the work done, to the nearest tenth.5303.3 ft⋅lb
  3. A worker pushes a lawn mower 20 ft with 20 lb along the handle, which is at 45∘ below the horizontal. Find the work done, to the nearest tenth.282.8 ft⋅lb
  4. A shopper pushes a cart 110 ft with 55 lb along a handle at 20∘ below the horizontal. Find the work done, to the nearest tenth.5685.1 ft⋅lb
  5. A child drags a sled 160 ft with a rope at 30∘ to the ground, pulling with 10 lb. Find the work done, to the nearest tenth.1385.6 ft⋅lb

  1. Decide whether vectors are orthogonal

    Are u⃗ and v⃗ orthogonal? u⃗=⟨−2,−3⟩, v⃗=⟨−3,−5⟩No
  2. Are u⃗ and v⃗ orthogonal? u⃗=⟨−9,5⟩, v⃗=⟨−5,−12⟩No

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

How to do these

The angle between ⟨3,4⟩ and ⟨5,12⟩

  1. The dot product is 15 plus 48, which is 63.
  2. The magnitudes are 5 and 13, so the product is 65.
  3. The angle is arccos of 63 over 65.

The answer is 14.3∘.

Where students go wrong

Expecting a vector back. A dot product is a number, so an answer written in angle parentheses is the wrong kind of thing. The cross product is the one that returns a vector.

Also called scalar product, angle between vectors, orthogonal vectors or vector projection.

What you can put on this worksheet

The dot product
Find u⃗⋅v⃗: u⃗=⟨1,1⟩, v⃗=⟨3,−6⟩ → −3
The angle between two vectors
Find the angle between u⃗ and v⃗: u⃗=⟨3,1⟩, v⃗=⟨3,−5⟩ → 77.5∘
Projections
Find the projection of u⃗ onto v⃗: u⃗=⟨7,9⟩, v⃗=⟨3,1⟩ → ⟨9,3⟩
Orthogonal or not
Are u⃗ and v⃗ orthogonal? u⃗=⟨4,1⟩, v⃗=⟨−1,6⟩ → No
Parallel, orthogonal, or neither
u⃗=⟨6,−4⟩, v⃗=⟨12,18⟩. Parallel, orthogonal, or neither? → Orthogonal
Work done by a force
A force of 60 lb at 40∘ to the horizontal pulls a wagon 120 ft. Find the work done, to the nearest tenth. → 5515.5 ft⋅lb
Read the vectors off a graph
Find the dot product of the two vectors. → −55

Questions about these worksheets

Yes. Every dot product 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 dot product 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 two 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 7: the dot product, the angle between two vectors, projections, orthogonal or not, parallel, orthogonal, or neither, work done by a force and read the vectors off a graph. Tick as many as you want and set how many of each, or let it spread them evenly.

Expecting a vector back. A dot product is a number, so an answer written in angle parentheses is the wrong kind of thing. The cross product is the one that returns a vector.