What is the cross product of two vectors?

The cross product (A × B) of two 3D vectors produces a third vector that is perpendicular to both A and B. For vectors A = (ax, ay, az) and B = (bx, by, bz), the cross product is ((ay·bz − az·by), (az·bx − ax·bz), (ax·by − ay·bx)). Unlike the dot product, which returns a scalar, the cross product returns a vector. It is only defined for 3-dimensional vectors.

What is the right-hand rule for cross products?

The right-hand rule determines the direction of the cross product vector. Point your right-hand fingers in the direction of vector A, then curl them toward vector B — your thumb points in the direction of A × B. If you reverse the order (B × A), the result points in the opposite direction: B × A = −(A × B). This anti-commutativity is a key property of the cross product.

When is the cross product equal to zero?

The cross product is zero (the zero vector) when A and B are parallel or anti-parallel — that is, when the angle between them is 0° or 180°. It is also zero if either input is the zero vector. Geometrically, the magnitude of A × B equals |A| × |B| × sin(θ), so when θ = 0° or 180°, sin(θ) = 0 and the result is zero. This calculator shows a warning when the result magnitude is effectively zero.

What is the difference between the cross product and the dot product?

The dot product (A · B) returns a scalar equal to |A| × |B| × cos(θ), measuring how much two vectors point in the same direction. The cross product (A × B) returns a vector perpendicular to both, with magnitude |A| × |B| × sin(θ), measuring the degree to which they point in different directions. The dot product is maximum when vectors are parallel; the cross product is maximum when they are perpendicular (90°).

What are the real-world uses of the cross product in physics and engineering?

The cross product appears throughout physics and engineering: torque (τ = r × F, where r is the position vector and F is the force), magnetic force on a charge (F = qv × B), angular momentum (L = r × p), and normal vectors for surface orientation in computer graphics. In structural engineering, it is used to find the moment of a force about a point. In robotics and game development, cross products compute rotation axes and surface normals.

What does the magnitude of the cross product represent geometrically?

The magnitude of A × B equals the area of the parallelogram formed by A and B as adjacent sides. If you divide by 2, you get the area of the triangle with those two vectors as sides. For example, if |A × B| = 10, then the parallelogram has area 10 square units and the triangle has area 5 square units. This geometric interpretation makes the cross product useful in computational geometry for area and volume calculations.

How is the cross product used in computer graphics?

In 3D computer graphics, the cross product is essential for computing surface normal vectors — the direction perpendicular to a triangle face. Normals determine how light reflects off a surface: a surface normal pointing directly at a light source appears brightest; one pointing away appears dark. By taking the cross product of two edges of a triangle, graphics engines compute the face normal. Normal maps (used in modern games to add surface detail without adding geometry) are essentially textures that store cross-product-derived directions per pixel.

Can the cross product be calculated for 2D vectors?

The cross product is strictly defined for 3D vectors, but a 2D variant is commonly used. For 2D vectors A = (ax, ay) and B = (bx, by), the "2D cross product" is the scalar ax·by − ay·bx. This value equals the z-component of the 3D cross product when both vectors have z = 0. It is used to determine if three points form a left turn or right turn (positive = left turn, negative = right turn), which is the basis of convex hull algorithms and polygon winding-order tests in 2D computational geometry.

Cross Product Calculator — 3D Vector A × B

Calculates the cross product vector, magnitude, and angle between two 3D vectors

What Is the Cross Product?

This cross product calculator computes A × B — a binary operation on two 3D vectors that returns a third vector perpendicular to both inputs. Given vectors A = (ax, ay, az) and B = (bx, by, bz), the cross product is:

A × B = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx)

The magnitude of the result equals |A| × |B| × sin(θ), where θ is the angle between the two input vectors. This means the cross product is largest when A and B are perpendicular (sin 90° = 1) and zero when they are parallel (sin 0° = 0). This calculator returns the full result vector, its magnitude, the magnitude of each input vector, and the angle between A and B — all computed from the i, j, k components you enter.

The Cross Product Formula and How to Apply It

The determinant expansion is the standard way to remember the cross product formula. Write A and B as rows of a 3×3 matrix with unit vectors i, j, k in the first row:

| i   j   k  |
| ax  ay  az |
| bx  by  bz |

A × B = i(ay·bz − az·by) − j(ax·bz − az·bx) + k(ax·by − ay·bx)

Note the sign alternation: +i, −j, +k. A common mistake is to forget the negative sign on the j component, which flips that component of the result. This calculator handles the formula exactly, so you can use it to verify hand calculations.

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Cross Product vs. Dot Product

The cross product and dot product are the two fundamental vector multiplication operations, and they measure different things:

Dot product (A · B) — returns a scalar. Equals |A| × |B| × cos(θ). Maximum when vectors point in the same direction (θ = 0°). Used to find the component of one vector along another, to check perpendicularity (dot product = 0 means 90°), and to calculate work in physics (W = F · d).
Cross product (A × B) — returns a vector. Magnitude equals |A| × |B| × sin(θ). Maximum when vectors are perpendicular (θ = 90°). Used to find normal vectors, compute torque and angular momentum, and determine surface orientation.

For many geometry problems you need both. This calculator shows the angle between the vectors (derived from the dot product formula) alongside the cross product, so you have the full picture. The Pythagorean theorem calculator is useful for checking vector magnitudes — the magnitude of a 3D vector (ax, ay, az) is √(ax² + ay² + az²), which is the 3D extension of the Pythagorean theorem.

Physical Applications of the Cross Product

The cross product is fundamental in mechanics and electromagnetism:

Torque — τ = r × F, where r is the position vector from the pivot to the point of force application and F is the force vector. The magnitude is |r| × |F| × sin(θ), and the direction is the axis of rotation (right-hand rule).
Magnetic force — F = q(v × B), where q is charge, v is velocity, and B is the magnetic field. A positive charge moving perpendicular to a magnetic field experiences maximum force; a charge moving parallel to B experiences no force.
Angular momentum — L = r × p = r × (m·v). The cross product relates the linear momentum of an orbiting body to its angular momentum vector, which points along the rotation axis.
Surface normals — in 3D computer graphics, the cross product of two edge vectors of a triangle gives the surface normal needed for lighting calculations.

Geometric Interpretation: Area of a Parallelogram

The magnitude of A × B is equal to the area of the parallelogram with A and B as adjacent sides. Halve that value to get the area of the triangle formed by those two vectors. This makes the cross product valuable in computational geometry for checking whether points are collinear (zero cross product), calculating polygon areas, and determining if a point lies inside a triangle. For rectangular volume problems, the cubic feet calculator extends this idea to three dimensions using length × width × height.

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The Right-Hand Rule

The direction of A × B is determined by the right-hand rule: point your right-hand fingers along A, curl them toward B through the smaller angle, and your thumb points in the direction of A × B. This means the cross product is anti-commutative: B × A = −(A × B). The result vector is always perpendicular to the plane containing A and B, which is why it is used to define rotation axes and surface normals in 3D space.

When A and B are parallel (or one is zero), the cross product is the zero vector and the right-hand rule is undefined — there is no unique perpendicular direction. The calculator displays a warning in this case.

Sources & References

Cross Product — Khan Academy — Khan Academy
Cross Product — Wikipedia — Wikipedia

Cross Product Calculator