Cross Product Calculator

Orientation is the extra information a 3D vector pair carries beyond length and angle. The cross product calculator turns two three-component vectors into the perpendicular vector that records that orientation. It is the calculation behind torque directions, surface normals in graphics, signed area in geometry, and many mechanics problems where the order of vectors cannot be ignored.

Definition and calculation

The calculator asks for the x, y, and z components of vector 1 and vector 2. It converts all six entries to numbers and rejects the calculation if any entry is not finite. The returned primary value is vector 1 cross vector 2, displayed in angle-bracket notation. The individual i, j, and k components are also listed, along with the magnitude.

The cross product is defined only in three-dimensional Euclidean vector space for this calculator’s purpose. It produces a vector perpendicular to the two inputs. Its direction follows the right-hand rule: point your index finger along vector 1, curl toward vector 2, and your thumb points in the direction of vector 1 cross vector 2. Reversing the order flips the vector.

The component formula can be remembered as the determinant expansion of a symbolic matrix whose first row is the basis vectors. This is why the middle component often surprises learners: the standard determinant expansion includes a minus sign, which is already reflected in the calculator’s formula.

Formula and variables

Let the input vectors be

$\mathbf{a} = \langle a_x, a_y, a_z \rangle$

and

$\mathbf{b} = \langle b_x, b_y, b_z \rangle$

The calculation determines

$\mathbf{a} \times \mathbf{b} = \left\langle a_y b_z - a_z b_y,\ a_z b_x - a_x b_z,\ a_x b_y - a_y b_x \right\rangle$

The magnitude is then

$\left\lVert \mathbf{a} \times \mathbf{b} \right\rVert = \sqrt{x^2 + y^2 + z^2}$

where $x$, $y$, and $z$ are the three computed cross-product components. Geometrically,

$\left\lVert \mathbf{a} \times \mathbf{b} \right\rVert = \left\lVert \mathbf{a} \right\rVert \left\lVert \mathbf{b} \right\rVert \sin\theta$

where $\theta$ is the angle between the vectors. Component units multiply: meters crossed with meters gives square meters; meters crossed with newtons gives newton meters.

The component view and the area view are two descriptions of the same object. The component formula is what a computer can evaluate directly from six numbers. The area formula explains scale: doubling one input vector doubles every component of the cross product and doubles the parallelogram area. If both vectors are doubled, the magnitude becomes four times as large.

Worked example: two non-axis vectors

Use vector 1 as $\mathbf{a} = \langle 2, -1, 3 \rangle$ and vector 2 as $\mathbf{b} = \langle 4, 0, -2 \rangle$. The first component is

$a_y b_z - a_z b_y = (-1)(-2) - (3)(0) = 2$

The second component is

$a_z b_x - a_x b_z = (3)(4) - (2)(-2) = 12 + 4 = 16$

The third component is

$a_x b_y - a_y b_x = (2)(0) - (-1)(4) = 4$

So the calculator’s vector result is

$\mathbf{a} \times \mathbf{b} = \langle 2, 16, 4 \rangle$

The magnitude is

$\sqrt{2^2 + 16^2 + 4^2} = \sqrt{276} = 16.613247\ldots$

The magnitude rounds to 16.6132. The calculator rounds each component to four decimals before display, so this example appears as 2.0000, 16.0000, and 4.0000, with magnitude 16.6132.

Interpretation and applications

In physics, torque is often written as a lever-arm vector crossed with a force vector. The result’s magnitude tells how strongly the force tends to rotate an object, while its direction identifies the rotation axis. In computer graphics, cross products create surface normals, which decide how light hits a polygon and which side of a triangle faces the camera. In analytic geometry, the magnitude gives the area of a parallelogram; half of it gives the area of the triangle with the same spanning vectors.

Related OverCalculator tools support the surrounding work. The matrix calculator is useful when vectors are part of a linear transformation. The relative velocity calculator often pairs vector components in motion problems. For rates, slopes, and simpler component relationships, the slope calculator can help with two-dimensional geometry, while the scientific notation calculator keeps very large or small component magnitudes readable.

A quick perpendicularity check is possible after any result: the dot product of the cross product with either original vector should be zero, apart from rounding. That property is why the operation is so valuable for normals and rotation axes.

Edge cases and common mistakes

Parallel vectors return the zero vector because $\sin\theta = 0$ when $\theta$ is 0 degrees or 180 degrees. A zero input vector also returns zero. These cases are not errors; they mean the spanned parallelogram has zero area and the perpendicular direction is not uniquely determined by the two inputs.

The most common arithmetic mistake is changing the sign of the middle component. The calculator uses $a_z b_x - a_x b_z$, which is equivalent to the determinant expansion after the negative sign on the j term is accounted for. Another mistake is expecting ordinary multiplication rules. Cross products are not commutative, and they do not exist as a vector-valued operation for arbitrary dimensions in the way this calculator presents them. Finally, keep units attached: a result that represents area, torque, or angular momentum should not be reported as if it had the same units as the original vectors.

Sources

• OpenStax, Calculus Volume 3: The Cross Product — component formula, right-hand rule, and geometric magnitude.
• Wolfram MathWorld, Cross Product — determinant form and algebraic properties of the vector product.