Cross Product Calculator

Find the oriented vector perpendicular to two 3D vectors, then check its magnitude and direction.

Drag & drop or upload an image or PDF

Enter two 3D vectors, such as cross([2,3,4],[1,-2,2])

Enter vectors in brackets, such as [1, 2, 3]. Keep both vectors in the same coordinate order.

Try an example

The cross product keeps direction and orientation

The standard cross product takes two 3D real vectors and returns a vector perpendicular to both. Its direction follows the right-hand rule, so reversing the input order reverses the sign.

The magnitude equals the area of the parallelogram spanned by the vectors. Half of that magnitude gives the corresponding triangle area. Parallel vectors have a zero cross product and no unique unit normal.

How to use the cross product calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Component form of the cross product

Compute the three signed component differences in order. Swapping the vectors negates the result.

a \times b = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1}

A reliable way to work through it

Keep the vector order visible

Write a × b before expanding because b × a points in the opposite direction.

Protect the middle sign

Use the component formula carefully so the second coordinate is not given the wrong sign.

Check perpendicularity

Dot the result with each input vector. Both checks should equal zero.

Worked example

Find a × b

a = 234, b = 1 - 2 2

Keep the first-cross-second order fixed.

a \times b = 3 (2) - 4 (- 2) 4 (1) - 2 (2) 2 (- 2) - 3 (1) = 140 - 7

Compute each signed component.

140 - 7 \cdot a = 0, 140 - 7 \cdot b = 0

Verify the result is perpendicular to both inputs.

∥ a \times b ∥ = 1 4^{2} + (- 7)^{2} = 75

The magnitude is the parallelogram area.

The cross product is [14, 0, -7], and its magnitude is 7√5.

Common mistakes to check

Reversing the order silently

Swapping the vectors negates the cross product, so the original orientation must stay visible.

Losing the middle sign

The second component is a₃b₁ - a₁b₃ in the direct component formula.

Returning a scalar

A cross product returns a perpendicular vector. A scalar result belongs to the dot product.

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Questions students ask

Is the cross product a scalar or a vector?

It is a vector perpendicular to both input vectors in the standard 3D setting.

How is its direction chosen?

Use the right-hand rule from the first vector toward the second.

Why can the cross product be zero?

The vectors are parallel, antiparallel, or one of them is the zero vector.

Can I use 2D vectors?

They can be embedded as [x, y, 0]. The result then has only a signed z component.