Cross Product CalculatorFree online math calculator — instant results, no sign-up
Calculate the cross product of two three-dimensional vectors.
About the Cross Product Calculator
The Cross Product Calculator calculates the vector perpendicular to two three-dimensional vectors by taking the first vector crossed with the second. Enter their x, y, and z components to obtain the three components of the cross product. With the default vectors (1, 0, 0) and (0, 1, 0), the result is (0, 0, 1).
- First x
- The x-component of the first vector, defaulting to 1, is expressed in the same units as its y- and z-components or is unitless for a purely numerical vector.
- First y
- The y-component of the first vector, defaulting to 0, is expressed in the same units as its x- and z-components or is unitless for a purely numerical vector.
- First z
- The z-component of the first vector, defaulting to 0, is expressed in the same units as its x- and y-components or is unitless for a purely numerical vector.
- Second x
- The x-component of the second vector, defaulting to 0, is expressed in the same units as its y- and z-components or is unitless for a purely numerical vector.
- Second y
- The y-component of the second vector, defaulting to 1, is expressed in the same units as its x- and z-components or is unitless for a purely numerical vector.
- Second z
- The z-component of the second vector, defaulting to 0, is expressed in the same units as its x- and y-components or is unitless for a purely numerical vector.
Formula and method
For a = (a_x, a_y, a_z) and b = (b_x, b_y, b_z), a × b = (x, y, z), where x = a_y b_z − a_z b_y, y = a_z b_x − a_x b_z, and z = a_x b_y − a_y b_x. With the defaults: x = 0·0 − 0·1 = 0; y = 0·0 − 1·0 = 0; z = 1·1 − 0·0 = 1.
- aThe first input vector.
- bThe second input vector.
- a_x, a_y, a_zThe First x, First y, and First z input values, respectively.
- b_x, b_y, b_zThe Second x, Second y, and Second z input values, respectively.
- ×The cross product operator, with operand order determining the result's direction.
- x, y, zThe three output components of the vector a × b.
- ·Ordinary multiplication between scalar components in the worked calculation.
The calculator forms each output component by subtracting two products of input components. It uses the y- and z-components for x, the z- and x-components for y, and the x- and y-components for z. The order of these subtractions implements the right-hand-rule orientation for the first vector crossed with the second; the result is not normalized.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- First x1
- First y0
- First z0
- Second x0
- Second y1
- Second z0
Results
- x0
- y0
- z1
How to use the Cross Product Calculator
- 1Enter the first vector in First x, First y, and First z; for the default example, use 1, 0, and 0.
- 2Enter the second vector in Second x, Second y, and Second z; for the default example, use 0, 1, and 0.
- 3Check that both vectors use the same Cartesian coordinate frame and that component units are consistent within each vector.
- 4Calculate the cross product in the order first vector × second vector.
- 5Read the output rows: the default inputs produce x = 0, y = 0, and z = 1, giving the vector (0, 0, 1).
Reading your result
The x, y, and z output rows are signed components of the resulting vector along the corresponding coordinate axes, not three separate magnitudes. For the defaults, x = 0 and y = 0 indicate no component along those axes, while z = 1 indicates a vector pointing along the positive z-axis. A nonzero result is perpendicular to both inputs, with direction given by the right-hand rule. Its magnitude equals the area of the parallelogram spanned by the input vectors. Output units are the product of the two vectors' units; unitless inputs give a unitless result.
Limitations
This calculator uses the standard three-dimensional cross product in a shared, right-handed orthonormal Cartesian coordinate frame. It does not convert units or coordinate systems, and it does not automatically produce a unit normal. Parallel or antiparallel inputs, or either input being the zero vector, produce a zero vector with no defined direction. Finite-precision arithmetic may introduce rounding error, especially when nearly parallel vectors cause subtraction of nearly equal products.
Common questions
Why do the default vectors produce a positive z-component?+
The first vector (1, 0, 0) points along positive x, and the second (0, 1, 0) points along positive y. The right-hand rule gives positive z, and the formula confirms z = 1·1 − 0·0 = 1 while x and y are both 0.
What happens if I swap the first and second vectors?+
The cross product changes sign: b × a = −(a × b). Swapping the default vectors therefore changes (0, 0, 1) to (0, 0, −1), without changing its magnitude.
What does a cross product of (0, 0, 0) mean?+
If both input vectors are nonzero, they are parallel or antiparallel. A zero result also occurs if either input is the zero vector. In either case, the inputs do not define a unique normal direction through their cross product.
Is the output always a unit normal vector?+
No. Its magnitude depends on the lengths of the input vectors and the angle between them. To obtain a unit normal, divide a nonzero result by its magnitude, √(x² + y² + z²); the default result (0, 0, 1) already has magnitude 1.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.