Worksheets · Precalculus

Cross product worksheet

The cross product of two vectors is a third vector, at right angles to both. Its length is the area of the parallelogram they span, which is what makes it an area tool as well as a direction tool.

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

Cross Products

Date Period

Find each answer.

  1. Find u⃗×v⃗: u⃗=⟨3,−12,1⟩, v⃗=⟨12,12,−5⟩
  2. Find the area of the parallelogram with sides u⃗=⟨1,3,−1⟩ and v⃗=⟨6,−1,0⟩
  3. Find u⃗×v⃗: u⃗=⟨−9,−2,−6⟩, v⃗=⟨−9,0,−11⟩
  4. Find the area of the parallelogram with sides u⃗=⟨−1,3,−3⟩ and v⃗=⟨−4,−6,−1⟩
  5. Find u⃗×v⃗: u⃗=⟨2,7,−5⟩, v⃗=⟨−7,4,1⟩
  6. Find the area of the parallelogram with sides u⃗=⟨3,−4,3⟩ and v⃗=⟨−1,4,6⟩
  7. Find u⃗×v⃗: u⃗=⟨−7,6,−9⟩, v⃗=⟨−4,−2,−6⟩
  8. Find the area of the parallelogram with sides u⃗=⟨−5,3,1⟩ and v⃗=⟨−5,−3,−6⟩
  9. Find u⃗×v⃗: u⃗=⟨3,9,−7⟩, v⃗=⟨12,5,−2⟩
  10. Find the area of the parallelogram with sides u⃗=⟨−3,2,6⟩ and v⃗=⟨6,3,5⟩
  11. Find u⃗×v⃗: u⃗=⟨8,11,−10⟩, v⃗=⟨−12,9,−5⟩
  12. Find the area of the parallelogram with sides u⃗=⟨5,0,6⟩ and v⃗=⟨3,−4,2⟩
  13. Find u⃗×v⃗: u⃗=⟨11,−9,12⟩, v⃗=⟨−8,−3,−10⟩
  14. Find the area of the parallelogram with sides u⃗=⟨−1,−2,2⟩ and v⃗=⟨0,0,−6⟩
  15. Find u⃗×v⃗: u⃗=⟨8,9,−6⟩, v⃗=⟨−8,−5,10⟩
  16. Find the area of the parallelogram with sides u⃗=⟨−3,−5,2⟩ and v⃗=⟨1,−2,−5⟩
  17. Find u⃗×v⃗: u⃗=⟨−9,−11,−5⟩, v⃗=⟨4,4,−10⟩
  18. Find the area of the parallelogram with sides u⃗=⟨0,0,−4⟩ and v⃗=⟨4,−3,6⟩
  19. Find u⃗×v⃗: u⃗=⟨11,9,−12⟩, v⃗=⟨−8,−12,12⟩
  20. Find the area of the parallelogram with sides u⃗=⟨−2,3,5⟩ and v⃗=⟨−5,−3,5⟩

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

Sigma Prep · sigmaprep.io/worksheets

Name

Cross ProductsAnswer keyVersion 1

Date Period

Find each answer.

  1. Find u⃗×v⃗: u⃗=⟨3,−12,1⟩, v⃗=⟨12,12,−5⟩⟨48,27,180⟩
  2. Find the area of the parallelogram with sides u⃗=⟨1,3,−1⟩ and v⃗=⟨6,−1,0⟩398
  3. Find u⃗×v⃗: u⃗=⟨−9,−2,−6⟩, v⃗=⟨−9,0,−11⟩⟨22,−45,−18⟩
  4. Find the area of the parallelogram with sides u⃗=⟨−1,3,−3⟩ and v⃗=⟨−4,−6,−1⟩886
  5. Find u⃗×v⃗: u⃗=⟨2,7,−5⟩, v⃗=⟨−7,4,1⟩⟨27,33,57⟩
  6. Find the area of the parallelogram with sides u⃗=⟨3,−4,3⟩ and v⃗=⟨−1,4,6⟩1801
  7. Find u⃗×v⃗: u⃗=⟨−7,6,−9⟩, v⃗=⟨−4,−2,−6⟩⟨−54,−6,38⟩
  8. Find the area of the parallelogram with sides u⃗=⟨−5,3,1⟩ and v⃗=⟨−5,−3,−6⟩594
  9. Find u⃗×v⃗: u⃗=⟨3,9,−7⟩, v⃗=⟨12,5,−2⟩⟨17,−78,−93⟩
  10. Find the area of the parallelogram with sides u⃗=⟨−3,2,6⟩ and v⃗=⟨6,3,5⟩3106
  11. Find u⃗×v⃗: u⃗=⟨8,11,−10⟩, v⃗=⟨−12,9,−5⟩⟨35,160,204⟩
  12. Find the area of the parallelogram with sides u⃗=⟨5,0,6⟩ and v⃗=⟨3,−4,2⟩465
  13. Find u⃗×v⃗: u⃗=⟨11,−9,12⟩, v⃗=⟨−8,−3,−10⟩⟨126,14,−105⟩
  14. Find the area of the parallelogram with sides u⃗=⟨−1,−2,2⟩ and v⃗=⟨0,0,−6⟩65
  15. Find u⃗×v⃗: u⃗=⟨8,9,−6⟩, v⃗=⟨−8,−5,10⟩⟨60,−32,32⟩
  16. Find the area of the parallelogram with sides u⃗=⟨−3,−5,2⟩ and v⃗=⟨1,−2,−5⟩1131
  17. Find u⃗×v⃗: u⃗=⟨−9,−11,−5⟩, v⃗=⟨4,4,−10⟩⟨130,−110,8⟩
  18. Find the area of the parallelogram with sides u⃗=⟨0,0,−4⟩ and v⃗=⟨4,−3,6⟩20
  19. Find u⃗×v⃗: u⃗=⟨11,9,−12⟩, v⃗=⟨−8,−12,12⟩⟨−36,−36,−60⟩
  20. Find the area of the parallelogram with sides u⃗=⟨−2,3,5⟩ and v⃗=⟨−5,−3,5⟩3174

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

How to do these

⟨1,2,3⟩×⟨4,5,6⟩

  1. First component: 2 times 6 minus 3 times 5.
  2. Second: 3 times 4 minus 1 times 6.
  3. Third: 1 times 5 minus 2 times 4.

The answer is ⟨−3,6,−3⟩.

Where students go wrong

Doing the middle component the same way round as the other two. It is the one that flips, so it reads z times the first x minus x times the first z. Getting that backwards puts a sign error in exactly one component, which is easy to miss.

Also called vector product, area of a parallelogram or orthogonal vector.

What you can put on this worksheet

The cross product
Find u⃗×v⃗: u⃗=⟨0,1,2⟩, v⃗=⟨−4,0,1⟩ → ⟨1,−8,4⟩
The area of a parallelogram
Find the area of the parallelogram with sides u⃗=⟨0,1,2⟩ and v⃗=⟨−4,0,1⟩ → 9
The area of a triangle in space
Find the area of the triangle A(0,1,2), B(−4,4,3), C(−2,5,1) → 1852
Parallel vectors
Find u⃗×v⃗: u⃗=⟨2,1,2⟩, v⃗=⟨−4,−2,−4⟩ → ⟨0,0,0⟩

Questions about these worksheets

Yes. Every cross 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 cross 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 4: the cross product, the area of a parallelogram, the area of a triangle in space and parallel vectors. Tick as many as you want and set how many of each, or let it spread them evenly.

Doing the middle component the same way round as the other two. It is the one that flips, so it reads z times the first x minus x times the first z. Getting that backwards puts a sign error in exactly one component, which is easy to miss.