Math
Cross Product Calculator
Enter two 3D vectors and mathcheck expands the i, j, k determinant term by term to get a × b.
It also gives the magnitude of the result (the area of the parallelogram the vectors span), the triangle area, the unit normal, the angle between the vectors, and verifies that the result is perpendicular to both inputs.
Try an example
Result
Cross product a × b
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- Magnitude |a × b| (parallelogram area)
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- Unit normal
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- Angle between a and b
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- Triangle area
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- b × a
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the cross product is
The cross product of two 3D A quantity with both a size and a direction, often drawn as an arrow. is a new vector perpendicular to both of them. Its direction follows the right-hand rule (curl your fingers from a toward b; your thumb points along a × b) and its length equals |a| |b| sin θ, which is the area of the parallelogram with sides a and b.
Using the determinant
Write the unit vectors i, j, k in the first row, the components of a in the second, and the components of b in the third. Expand along the first row: the i term uses the 2×2 One number computed from a square matrix that tells you whether it can be inverted. of the y and z columns, the j term uses the x and z columns with a minus sign, and the k term uses the x and y columns.
⟨1, 2, 3⟩ × ⟨4, 5, 6⟩
i: 2·6 - 3·5 = -3. j: 3·4 - 1·6 = 6. k: 1·5 - 2·4 = -3. So a × b = ⟨-3, 6, -3⟩ with magnitude sqrt(9 + 36 + 9) = 3√6 ≈ 7.35. Check: ⟨-3, 6, -3⟩ · ⟨1, 2, 3⟩ = -3 + 12 - 9 = 0.
How to use this calculator
- Enter the three components of vector a and vector b.
- Read a × b, then its magnitude, which doubles as the parallelogram area.
- Use the unit normal when you need a direction perpendicular to both vectors, for example a plane's normal.
- Check b × a if your problem uses the opposite order.
- Open Show the work for the determinant expansion and the perpendicularity check.
How to read the answer
The result is perpendicular to both inputs, and the calculator confirms this by showing that its Multiply two vectors piece by piece and add the results to get one number. with each input is 0. A zero cross product means the vectors are parallel, so they do not define a plane. The The length or size of a vector, ignoring which way it points. tells you the parallelogram area; halve it for the triangle with the same two sides.
Common mistakes and edge cases
- Forgetting the minus sign on the j component when expanding the determinant.
- Swapping a and b, which negates the answer.
- Trying to cross 2D vectors. Treat them as ⟨x, y, 0⟩; the result then points along the z-axis.
- Confusing the cross product (a vector) with the dot product (a number).
- Parallel vectors give the zero vector, not an error; there is simply no unit normal.
Worked examples
Textbook example
⟨1, 2, 3⟩ × ⟨4, 5, 6⟩
Cross product a × b
⟨-3, 6, -3⟩
Standard basis
i × j = k
Cross product a × b
⟨0, 0, 1⟩
Reversed order
j × i = -k
Cross product a × b
⟨0, 0, -1⟩
Parallel vectors
⟨2, 4, 6⟩ × ⟨1, 2, 3⟩
Cross product a × b
⟨0, 0, 0⟩
2D vectors embedded in 3D
⟨3, 4, 0⟩ × ⟨-1, 2, 0⟩
Cross product a × b
⟨0, 0, 10⟩
Decimal components
⟨0.5, 1, 0⟩ × ⟨0, 2, 1.5⟩
Cross product a × b
⟨1.5, -0.75, 1⟩
Frequently asked questions
How do you calculate a cross product?+
Use a × b = ⟨a2·b3 - a3·b2, a3·b1 - a1·b3, a1·b2 - a2·b1⟩, or expand the 3×3 determinant with i, j, k in the top row.
What does the cross product represent?+
A vector perpendicular to both inputs whose length equals the area of the parallelogram they span. In physics it appears in torque, angular momentum, and the magnetic force.
Why is the cross product zero for parallel vectors?+
Because |a × b| = |a| |b| sin θ and sin 0° = sin 180° = 0. Parallel vectors do not enclose any area.
Does the order of the vectors matter?+
Yes. b × a = -(a × b). The magnitude is the same but the direction is reversed, following the right-hand rule.
Can I take the cross product of 2D vectors?+
Only by treating them as 3D vectors with z = 0. The result is ⟨0, 0, a1·b2 - a2·b1⟩, and that z-value is the signed parallelogram area.
How do I find a normal vector to a plane?+
Take two non-parallel vectors lying in the plane and cross them. Divide by the magnitude if you need a unit normal.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026