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. Dot it with a third vector and the absolute value is the volume of the parallelepiped the three span.

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Sigma Prep · sigmaprep.io/worksheets

Name

Cross Products

Date Period

Find each answer.

  1. Find the cross product

    Find u⃗×v⃗: u⃗=⟨1,−4,0⟩, v⃗=⟨4,4,−2⟩
  2. Find u⃗×v⃗: u⃗=⟨−8,−11,−2⟩, v⃗=⟨2,7,−5⟩
  3. Find u⃗×v⃗: u⃗=⟨4,2,−1⟩, v⃗=⟨−2,2,4⟩
  4. Find u⃗×v⃗: u⃗=⟨11,−9,12⟩, v⃗=⟨−8,−3,−10⟩
  5. Find u⃗×v⃗: u⃗=⟨0,−2,−3⟩, v⃗=⟨−3,−4,−2⟩
  6. Find u⃗×v⃗: u⃗=⟨−4,6,9⟩, v⃗=⟨−10,−6,9⟩
  7. Find u⃗×v⃗: u⃗=⟨−1,−2,−3⟩, v⃗=⟨−2,4,4⟩

  1. Find the area of a parallelogram

    Find the area of the parallelogram with sides u⃗=⟨1,3,−1⟩ and v⃗=⟨6,−1,0⟩
  2. Find the area of the parallelogram with sides u⃗=⟨−3,1,0⟩ and v⃗=⟨2,−3,2⟩
  3. Find the area of the parallelogram with sides u⃗=⟨6,3,5⟩ and v⃗=⟨4,6,−5⟩
  4. Find the area of the parallelogram with sides u⃗=⟨−1,−2,1⟩ and v⃗=⟨0,0,−4⟩
  5. Find the area of the parallelogram with sides u⃗=⟨2,2,−5⟩ and v⃗=⟨0,0,−4⟩
  6. Find the area of the parallelogram with sides u⃗=⟨3,3,−3⟩ and v⃗=⟨−4,−3,−1⟩
  7. Find the area of the parallelogram with sides u⃗=⟨−4,−2,−6⟩ and v⃗=⟨−6,4,−1⟩

  1. Find the volume of a parallelepiped

    Find the volume of the parallelepiped with edges u⃗=⟨−2,0,−2⟩, v⃗=⟨−2,0,−3⟩, w⃗=⟨0,3,−1⟩.
  2. Find the volume of the parallelepiped with edges u⃗=⟨−2,2,1⟩, v⃗=⟨−2,−1,−3⟩, w⃗=⟨1,3,−2⟩.
  3. Find the volume of the parallelepiped with edges u⃗=⟨−3,3,−1⟩, v⃗=⟨4,1,4⟩, w⃗=⟨2,−2,2⟩.
  4. Find the volume of the parallelepiped with edges u⃗=⟨3,3,−1⟩, v⃗=⟨−2,−1,4⟩, w⃗=⟨−1,−2,2⟩.
  5. Find the volume of the parallelepiped with edges u⃗=⟨3,−1,4⟩, v⃗=⟨4,3,−3⟩, w⃗=⟨−2,−3,4⟩.
  6. Find the volume of the parallelepiped with edges u⃗=⟨1,−1,4⟩, v⃗=⟨−1,4,3⟩, w⃗=⟨2,2,−2⟩.

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 the cross product

    Find u⃗×v⃗: u⃗=⟨1,−4,0⟩, v⃗=⟨4,4,−2⟩⟨8,2,20⟩
  2. Find u⃗×v⃗: u⃗=⟨−8,−11,−2⟩, v⃗=⟨2,7,−5⟩⟨69,−44,−34⟩
  3. Find u⃗×v⃗: u⃗=⟨4,2,−1⟩, v⃗=⟨−2,2,4⟩⟨10,−14,12⟩
  4. Find u⃗×v⃗: u⃗=⟨11,−9,12⟩, v⃗=⟨−8,−3,−10⟩⟨126,14,−105⟩
  5. Find u⃗×v⃗: u⃗=⟨0,−2,−3⟩, v⃗=⟨−3,−4,−2⟩⟨−8,9,−6⟩
  6. Find u⃗×v⃗: u⃗=⟨−4,6,9⟩, v⃗=⟨−10,−6,9⟩⟨108,−54,84⟩
  7. Find u⃗×v⃗: u⃗=⟨−1,−2,−3⟩, v⃗=⟨−2,4,4⟩⟨4,10,−8⟩

  1. Find the area of a parallelogram

    Find the area of the parallelogram with sides u⃗=⟨1,3,−1⟩ and v⃗=⟨6,−1,0⟩398
  2. Find the area of the parallelogram with sides u⃗=⟨−3,1,0⟩ and v⃗=⟨2,−3,2⟩89
  3. Find the area of the parallelogram with sides u⃗=⟨6,3,5⟩ and v⃗=⟨4,6,−5⟩5101
  4. Find the area of the parallelogram with sides u⃗=⟨−1,−2,1⟩ and v⃗=⟨0,0,−4⟩45
  5. Find the area of the parallelogram with sides u⃗=⟨2,2,−5⟩ and v⃗=⟨0,0,−4⟩82
  6. Find the area of the parallelogram with sides u⃗=⟨3,3,−3⟩ and v⃗=⟨−4,−3,−1⟩342
  7. Find the area of the parallelogram with sides u⃗=⟨−4,−2,−6⟩ and v⃗=⟨−6,4,−1⟩669

  1. Find the volume of a parallelepiped

    Find the volume of the parallelepiped with edges u⃗=⟨−2,0,−2⟩, v⃗=⟨−2,0,−3⟩, w⃗=⟨0,3,−1⟩.6
  2. Find the volume of the parallelepiped with edges u⃗=⟨−2,2,1⟩, v⃗=⟨−2,−1,−3⟩, w⃗=⟨1,3,−2⟩.41
  3. Find the volume of the parallelepiped with edges u⃗=⟨−3,3,−1⟩, v⃗=⟨4,1,4⟩, w⃗=⟨2,−2,2⟩.20
  4. Find the volume of the parallelepiped with edges u⃗=⟨3,3,−1⟩, v⃗=⟨−2,−1,4⟩, w⃗=⟨−1,−2,2⟩.15
  5. Find the volume of the parallelepiped with edges u⃗=⟨3,−1,4⟩, v⃗=⟨4,3,−3⟩, w⃗=⟨−2,−3,4⟩.5
  6. Find the volume of the parallelepiped with edges u⃗=⟨1,−1,4⟩, v⃗=⟨−1,4,3⟩, w⃗=⟨2,2,−2⟩.58

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 the first z times the second x minus the first x times the second z. Getting that backwards puts a sign error in exactly one component, which is easy to miss.

Also called vector product 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⟩
Volume of a parallelepiped
Find the volume of the parallelepiped with edges u⃗=⟨1,1,2⟩, v⃗=⟨−3,0,1⟩, w⃗=⟨−2,4,−1⟩. → 33
From four points: PQ x RS
Given P(0,0,2), Q(−3,0,0), R(−2,3,−1), S(2,0,3), find PQ→×RS→ → PQ→=⟨−3,0,−2⟩, RS→=⟨4,−3,4⟩; ⟨−6,4,9⟩
A vector orthogonal to both
Find a vector orthogonal to both u=⟨0,0,2⟩ and v=⟨−3,0,0⟩ → ⟨0,−1,0⟩ (or any nonzero multiple)
Parallelogram area from three points
Find the area of the parallelogram with adjacent sides AB→ and AC→: A(0,0,2), B(−3,0,0), C(−2,3,−1) → 142
Parallelepiped volume from four vertices
Find the volume of the parallelepiped with edges TX→,TY→,TZ→: T(0,0,2), X(−3,0,0), Y(−2,3,−1), Z(2,0,3) → 3

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 9: the cross product, the area of a parallelogram, the area of a triangle in space, parallel vectors, volume of a parallelepiped, from four points: pq x rs, a vector orthogonal to both, parallelogram area from three points and parallelepiped volume from four vertices. 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 the first z times the second x minus the first x times the second z. Getting that backwards puts a sign error in exactly one component, which is easy to miss.