Cross Product Calculator

Compute perpendicular vectors, magnitudes, and geometric area from two vectors.

Results are calculated automatically as you enter data.

Vector dimension

Vector A x y z

Vector B x y z

Determinant expansion

A cross B Enter valid vectors.

Rotate Horizontal 45°

Rotate Vertical 30°

Disclaimer: All tools on this site are intended for educational and informational purposes only. Not professional advice. No guarantee of accuracy; verify units, formulas, assumptions, tolerances, rounding, and limitations before important use. See our Terms of Use.

Assumptions and limitations

Results are educational estimates, not professional engineering, construction, safety, scientific, legal, financial, or academic advice. No guarantee is made that outputs are accurate, complete, suitable, or error-free. Verify units, formulas, assumptions, tolerances, rounding, and limitations independently, and use qualified professional review for technical or safety-sensitive work. Use of this calculator is subject to our Terms of Use.

▼ See explanations and tips below ▼

Related Calculators

What Is the Cross Product?

The cross product is a way to multiply two vectors so that the result is a new vector. Unlike the dot product, which returns a single number, the cross product returns a vector that is perpendicular to the two input vectors.

The most common setting for the cross product is three-dimensional Cartesian space. If vector $\mathbf{A}$ and vector $\mathbf{B}$ both lie in a plane, then $\mathbf{A} \times \mathbf{B}$ points in one of the two directions perpendicular to that plane. The right-hand rule decides which direction is positive.

The cross product also has a geometric meaning: its magnitude is the area of the parallelogram formed by the two vectors. Because a triangle made from the same two side vectors has half that area, the cross product is also a useful shortcut for triangle area.

In two dimensions, a common practical shortcut is to treat both vectors as if their $z$-components are $0$. The result points along the $z$-axis, so it is often reported as just the signed $k$ component. That sign tells orientation: positive in the standard counterclockwise direction and negative in the clockwise direction.

Why the Cross Product Matters

The cross product connects algebra, geometry, and physical direction in one operation. It helps answer questions such as:

What direction is perpendicular to two given vectors?
What is the area spanned by two side vectors?
Are two vectors parallel?
Is one 2D vector clockwise or counterclockwise from another?
What is the direction of a torque, angular momentum vector, or surface normal?

Students usually meet the cross product in vector algebra, calculus, and physics. It also appears in engineering, computer graphics, robotics, geometry, and mechanics because perpendicular directions and oriented areas are common in those fields.

Key Terms to Know

Vector: A quantity with components, often used to represent direction and magnitude.
Component: One coordinate value of a vector, such as $a_x$, $a_y$, or $a_z$.
Cartesian coordinates: Coordinates measured along perpendicular axes such as $x$, $y$, and $z$.
Magnitude: The length of a vector, written as $\lVert \mathbf{A} \rVert$.
Determinant: A structured calculation that can represent area, volume, orientation, and the component formula for a cross product.
Right-hand rule: A convention for choosing the direction of $\mathbf{A} \times \mathbf{B}$.
Normal vector: A vector perpendicular to a line, plane, or surface.
Dot product: A vector operation that returns a scalar and is useful for measuring angles and projection.

How the Cross Product Works

Let the two 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 cross product is

$$ \mathbf{A} \times \mathbf{B} = \left\langle a_yb_z - a_zb_y, a_zb_x - a_xb_z, a_xb_y - a_yb_x \right\rangle. $$

A compact way to remember the same formula is the determinant pattern

$$ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}. $$

The result has three components:

The $x$ component is $a_yb_z - a_zb_y$.
The $y$ component is $a_zb_x - a_xb_z$.
The $z$ component is $a_xb_y - a_yb_x$.

The geometric size of the cross product is

$$ \lVert \mathbf{A} \times \mathbf{B} \rVert = \lVert \mathbf{A} \rVert\,\lVert \mathbf{B} \rVert\sin\theta, $$

where $\theta$ is the angle between the two vectors. This value is also the area of the parallelogram spanned by $\mathbf{A}$ and $\mathbf{B}$:

$$ \text{Parallelogram area} = \lVert \mathbf{A} \times \mathbf{B} \rVert. $$

The triangle with the same two side vectors has half that area:

$$ \text{Triangle area} = \frac{1}{2}\lVert \mathbf{A} \times \mathbf{B} \rVert. $$

The angle can be found by combining the cross product magnitude with the dot product:

$$ \theta = \operatorname{atan2}\left(\lVert \mathbf{A} \times \mathbf{B} \rVert,\, \mathbf{A} \cdot \mathbf{B}\right). $$

This works because $\lVert \mathbf{A} \times \mathbf{B} \rVert$ contains the sine part of the angle relationship, while $\mathbf{A} \cdot \mathbf{B}$ contains the cosine part.

The 2D Special Case

For 2D vectors, use $a_z = 0$ and $b_z = 0$. If

$$ \mathbf{A} = \langle a_x, a_y \rangle $$

and

$$ \mathbf{B} = \langle b_x, b_y \rangle, $$

then the cross product points only in the $k$ direction:

$$ \mathbf{A} \times \mathbf{B} = \langle 0, 0, a_xb_y - a_yb_x \rangle. $$

So the useful signed 2D value is

$$ k = a_xb_y - a_yb_x. $$

A positive $k$ means the turn from $\mathbf{A}$ to $\mathbf{B}$ is counterclockwise in the standard $xy$-plane. A negative $k$ means it is clockwise. A value of $0$ means the vectors are parallel or at least one vector is zero.

Examples of the Cross Product in Practice

Example 1: Simple 3D Cross Product

Suppose

$$ \mathbf{A} = \langle 2, -1, 3 \rangle $$

and

$$ \mathbf{B} = \langle 4, 0, -2 \rangle. $$

Use the component formula:

$$ \mathbf{A} \times \mathbf{B} = \left\langle (-1)(-2) - (3)(0), (3)(4) - (2)(-2), (2)(0) - (-1)(4) \right\rangle. $$

That gives

$$ \mathbf{A} \times \mathbf{B} = \langle 2, 16, 4 \rangle. $$

The magnitude is

$$ \lVert \mathbf{A} \times \mathbf{B} \rVert = \sqrt{2^2 + 16^2 + 4^2} = \sqrt{276} \approx 16.61324773. $$

So the parallelogram area is about $16.61324773$, and the triangle area is

$$ \frac{1}{2}\sqrt{276} \approx 8.30662386. $$

The dot product is

$$ \mathbf{A} \cdot \mathbf{B} = (2)(4) + (-1)(0) + (3)(-2) = 2. $$

So the angle is

$$ \theta = \operatorname{atan2}(\sqrt{276}, 2) \approx 83.13543^\circ. $$

Example 2: 2D Area and Orientation

Suppose two side vectors in a plane are

$$ \mathbf{A} = \langle 5, 0 \rangle $$

and

$$ \mathbf{B} = \langle 2, 3 \rangle. $$

The signed $k$ component is

$$ k = a_xb_y - a_yb_x = (5)(3) - (0)(2) = 15. $$

Because $k$ is positive, the turn from $\mathbf{A}$ to $\mathbf{B}$ is counterclockwise in the standard coordinate plane. The parallelogram area is

$$ |k| = 15. $$

The triangle area is

$$ \frac{|k|}{2} = \frac{15}{2} = 7.5. $$

This is why the 2D cross-product shortcut is useful in coordinate geometry: it gives both area and orientation from the same calculation.

Example 3: Common Edge Case With a Zero Vector

Suppose

$$ \mathbf{A} = \langle 0, 0, 0 \rangle $$

and

$$ \mathbf{B} = \langle 2, 1, 3 \rangle. $$

Every component of the cross product is $0$:

$$ \mathbf{A} \times \mathbf{B} = \langle 0, 0, 0 \rangle. $$

The parallelogram area is also $0$:

$$ \lVert \mathbf{A} \times \mathbf{B} \rVert = 0. $$

However, the angle between the vectors is not defined because the zero vector has no direction. This is different from two nonzero parallel vectors, where the angle is either $0^\circ$ or $180^\circ$ depending on whether they point the same way or opposite ways.

How to Interpret the Result

The cross product result has both direction and size.

In 3D, $\mathbf{A} \times \mathbf{B}$ points perpendicular to both input vectors. Use the right-hand rule to interpret the direction: point your right-hand fingers along $\mathbf{A}$ and curl them toward $\mathbf{B}$; your thumb points in the direction of $\mathbf{A} \times \mathbf{B}$.

The magnitude $\lVert \mathbf{A} \times \mathbf{B} \rVert$ tells how much area the two vectors span. A larger magnitude means a larger parallelogram area. A smaller magnitude means a smaller area. A magnitude of $0$ means the vectors do not span an area because they are parallel, antiparallel, or one of them is zero.

In 2D mode, the result is usually shown as the signed $k$ component. The absolute value gives the parallelogram area, while the sign gives orientation.

The angle result needs two nonzero vectors. When one vector is zero, there is no meaningful direction to compare, so the angle between the vectors is undefined.

If the vector components represent physical quantities, the units of the cross product come from multiplying the units of the two inputs. For example, position crossed with force gives torque units such as newton-meters. For pure coordinate vectors with no physical units, the results are unitless.

Common Mistakes and Misconceptions

A common mistake is reversing the order of the vectors. Cross product order matters:

$$ \mathbf{A} \times \mathbf{B} = -\left(\mathbf{B} \times \mathbf{A}\right). $$

The magnitude stays the same when the order is reversed, but the direction changes to the opposite direction.

Another mistake is expecting the cross product to behave like the dot product. The dot product returns a scalar, while the cross product returns a vector in 3D. In 2D, the signed $k$ component may look like a scalar, but it represents the $z$ component of a 3D cross product after setting both $z$ values to $0$.

Users also sometimes interpret a zero cross product too narrowly. A zero result can mean the vectors are parallel, but it can also happen when one or both input vectors are zero.

Another misconception is that the cross-product magnitude is only a vector length. It is a length, but it is also the area of the parallelogram formed by the two input vectors.

Finally, be careful with numeric input. Decimal values such as $0.5$ are valid numeric components, but expressions such as $1/2$ may not be accepted in calculator input fields that expect a plain number.

When to Use the Cross Product

Use the cross product when you need a perpendicular direction, an oriented area, or a vector quantity defined by rotation around an axis.

Common use cases include:

Finding a normal vector to a plane or surface.
Computing the area of a parallelogram or triangle from side vectors.
Determining clockwise or counterclockwise orientation in 2D geometry.
Checking whether two vectors are parallel.
Computing torque, moment, or angular momentum direction in physics and engineering.
Building surface normals in computer graphics and 3D modeling.
Understanding vector-valued formulas in multivariable calculus.

If you only need the angle or projection between two vectors, the dot product may be the better tool. If you need a perpendicular vector or area, the cross product is usually the better choice.

Limitations and Things to Keep in Mind

The standard cross product is a three-dimensional vector operation. A 2D cross-product value is best understood as the $k$ component produced after embedding the two vectors in the $xy$-plane with $z = 0$.

The calculation assumes Cartesian components. It does not directly handle polar, cylindrical, spherical, or other coordinate systems unless those coordinates are first converted into Cartesian components.

The calculator uses finite numeric inputs. It does not evaluate symbolic variables, vector strings such as $(1, 2, 3)$, fractions typed as expressions such as $1/2$, infinity, or undefined values. A nonzero decimal or scientific-notation input that is below the supported numeric range is rejected instead of being treated as a zero component.

The calculator rescales finite vectors internally before comparing their cross-product magnitude and dot product, so an angle such as $90^\circ$ is retained even when the final area is too small or too large for an ordinary binary floating-point value. Such results are shown in scientific notation when needed. A displayed $0$ is an exactly zero representable result, not a scale-relative cutoff that changes a nonzero area into zero.

The calculator does not perform unit conversion or dimensional analysis. If your vector components have physical units, you need to track the resulting units yourself.

The 3D visualization is a helpful geometric aid, but it is still a projection of a 3D situation onto a screen. Use the numeric vector result and the determinant steps as the main calculation.

How to Use This Calculator

Choose 2D or 3D mode.
Enter the $x$ and $y$ components for vector $\mathbf{A}$ and vector $\mathbf{B}$.
In 3D mode, also enter the $z$ components for both vectors.
Review $\mathbf{A} \times \mathbf{B}$, the magnitude, parallelogram area, triangle area, and angle between the vectors.
Use the determinant steps to check how each component was calculated.
In 3D mode, adjust the horizontal and vertical view angles to inspect the visualization.
Reset the view when you want to return to the default orientation.
Download the graph as a PNG when you need to save the current visualization.

Frequently Asked Questions

Is the cross product the same as the dot product?

No. The dot product returns a scalar, while the cross product returns a vector in 3D. The dot product is most useful for projection and angle relationships; the cross product is most useful for perpendicular direction and area.

Why does $\mathbf{A} \times \mathbf{B}$ point in the opposite direction from $\mathbf{B} \times \mathbf{A}$?

The cross product depends on the order of the two vectors. Reversing the order reverses the orientation, so the resulting vector points in the opposite direction even though its magnitude is the same.

What does the cross product mean in 2D?

In 2D, the vectors are treated as if their $z$ components are $0$. The result has only a $k$ component, $a_xb_y - a_yb_x$, which gives signed area and orientation.

What does a zero cross product mean?

A zero cross product means the two vectors do not span a parallelogram with positive area. This happens when the vectors are parallel, antiparallel, or when at least one vector is zero.

How is triangle area related to the cross product?

The magnitude of the cross product gives the area of the parallelogram spanned by two vectors. A triangle formed by the same two side vectors is half of that parallelogram, so its area is $\frac{1}{2}\lVert \mathbf{A} \times \mathbf{B} \rVert$.

Can cross product inputs have physical units?

Yes, the mathematical operation can be applied to components that represent physical quantities, but the calculator does not track or convert units. The resulting units come from multiplying the input units, such as meters times newtons for torque.

Sources and References

Books and Open Textbooks

Gilbert Strang and Edwin “Jed” Herman. Calculus Volume 3. OpenStax, 2016. Chapter 2, Sections 2.3 “The Dot Product” and 2.4 “The Cross Product.” https://openstax.org/books/calculus-volume-3/pages/2-3-the-dot-product and https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product
William Moebs, Samuel J. Ling, and Jeff Sanny. University Physics Volume 1. OpenStax, 2016. Chapter 2, Section 2.4 “Products of Vectors,” and Chapter 10, Section 10.6 “Torque.” https://openstax.org/books/university-physics-volume-1/pages/2-4-products-of-vectors and https://openstax.org/books/university-physics-volume-1/pages/10-6-torque

Online and Educational Sources

MIT OpenCourseWare. “Session 7: Cross Products.” 18.02SC Multivariable Calculus, Massachusetts Institute of Technology, Fall 2010. https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/pages/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-7-cross-products/
MIT OpenCourseWare. “Cross Product” course notes. 18.02SC Multivariable Calculus, Massachusetts Institute of Technology, Fall 2010. https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/38b6892f5294ed562cf3dedaf5b99ba4MIT1802SCnotes3.pdf
TU Delft Interactive Textbooks. “5.1. Determinants as areas or volumes.” Linear Algebra, accessed June 28, 2026. https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html