Cross Product Calculator — Vector Product of Two 3D Vectors with the Right-Hand Rule

Without a vector cross product calculator, it is hard to know how to calculate the cross product. Luckily for you, we've made a tool that helps you understand the formula for the cross product of two vectors. We will also be comparing the dot product vs. cross product definitions and explain why they are not the same operation. And as a bonus, we also have a list of practical tricks like the right-hand rule so that you can become a master of how to do the cross product of two vectors.

Vector cross product definition

A vector is a mathematical tool widely used in physics. It allows you to deal with collections of numbers (each representing a dimension) in a very efficient way. The collection of operations, rules, and properties to deal with vectors is called vector algebra and, similarly to the algebra of numbers, it includes multiplication.

However, vectors are more complex than numbers since they carry within them much more information that must be more carefully manipulated. This is one of the reasons why, in vector algebra, there are two different types of multiplications or product operations: the cross product and the dot product.

One definition of the cross product, also called the vector product, is:

A binary operation on two vectors in three-dimensional space that is denoted by the symbol ×. Given two linearly independent vectors, a and b, the cross product, a × b, is a vector perpendicular to both a and b and thus normal to the plane containing them.

In plain terms: the cross product takes two 3D vectors and returns a brand-new vector that sticks out at a right angle to both of them. That's why the cross product is only defined in three-dimensional space — you need "room" for the result to point out of the plane of the two input vectors.

Cross product formula

Given two 3D vectors a⃗ = (a_x, a_y, a_z) and b⃗ = (b_x, b_y, b_z), their cross product is the vector:

a⨯b = (a_yb_z − a_zb_y,   a_zb_x − a_xb_z,   a_xb_y − a_yb_x)

The magnitude of the cross product has its own tidy formula: |a⨯b| = |a| · |b| · sin(θ), where θ is the angle between the two vectors. This magnitude equals the area of the parallelogram spanned by a and b — and half of it is the area of the triangle they form.

How to do the cross product of two vectors

Let a⃗ = (1, 2, 3) and b⃗ = (4, 5, 6). Follow the three component formulas:

1. c_x = a_yb_z − a_zb_y = 2×6 − 3×5 = 12 − 15 = −3
2. c_y = a_zb_x − a_xb_z = 3×4 − 1×6 = 12 − 6 = 6
3. c_z = a_xb_y − a_yb_x = 1×5 − 2×4 = 5 − 8 = −3

So a⨯b = (−3, 6, −3), with magnitude |a⨯b| = √((−3)² + 6² + (−3)²) = √54 ≈ 7.3485. A handy memory aid is the determinant of a 3×3 matrix whose first row holds the unit vectors i, j, k, the second row is a, and the third row is b.

How to use the vector cross product calculator

1. Pick your unit system — metric (mm, cm, m, km) or US/Imperial (in, ft, yd, mi). The choice only affects how the components and the resulting area are labelled.
2. Enter the three components of vector a: a_x, a_y, a_z.
3. Enter the three components of vector b: b_x, b_y, b_z.
4. Read off the cross product a⨯b, its magnitude, the reverse product b⨯a, the dot product, the angle between the vectors, and the parallelogram and triangle areas — plus a step-by-step solution.

Dot product vs cross product

These two operations are easy to confuse, but they answer completely different questions:

• The dot product (a⋅b = a_xb_x + a_yb_y + a_zb_z) returns a single number (a scalar). It measures how much the two vectors point in the same direction and equals |a|·|b|·cos(θ).
• The cross product (a⨯b) returns a vector perpendicular to both inputs. Its length measures how much the vectors are perpendicular and equals |a|·|b|·sin(θ).

Two more differences worth remembering: the dot product is commutative (a⋅b = b⋅a), while the cross product is anti-commutative (a⨯b = −b⨯a). And the dot product works in any number of dimensions, while the cross product is special to three dimensions.

Cross product and physics: Best Friends Forever

The cross product shows up all over physics whenever a quantity depends on two directions and points "sideways" to both. The torque on a lever is τ = r × F. The magnetic part of the Lorentz force on a moving charge is F = qv × B. Angular momentum is L = r × p. In each case the cross product automatically produces a result perpendicular to the plane of the two inputs, exactly as the physics demands.

Right-hand rule in physics: why is it so useful?

Because a⨯b is perpendicular to the plane of a and b, there are two possible directions it could point — up or down. The right-hand rule settles the ambiguity: point the fingers of your right hand along a, curl them toward b, and your thumb points in the direction of a⨯b. This simple gesture lets you predict the direction of torques, magnetic forces, and rotational quantities without any calculation, which is why physicists and engineers reach for it constantly.

Metric vs. US (Imperial) Units

The cross product itself is purely geometric, so the numbers are the same whatever units you use. The unit selector only affects how the components and areas are labelled:

• Metric system: millimeters (mm), centimeters (cm), meters (m), kilometers (km). The magnitude and areas are reported in the squared unit (e.g. m²).
• US / Imperial system: inches (in), feet (ft), yards (yd), miles (mi), with areas in the squared unit (e.g. ft²).

FAQs

What is the cross product of two vectors?

It is a third vector, a⨯b, that is perpendicular to both input vectors. Its length equals |a|·|b|·sin(θ), the area of the parallelogram the vectors span.

Why is the cross product only defined in 3D?

The result must be perpendicular to both inputs. In three dimensions there is exactly one such direction (up to sign), which the right-hand rule fixes. In 2D there is no room for a perpendicular vector, and in higher dimensions the perpendicular direction is no longer unique.

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

The dot product returns a scalar and measures parallelism (|a|·|b|·cos θ); the cross product returns a perpendicular vector and measures perpendicularity (|a|·|b|·sin θ).

What happens if the two vectors are parallel?

Then sin(θ) = 0, so the cross product is the zero vector (0, 0, 0). Parallel vectors span no area, which matches the geometric meaning of |a⨯b|.

Is the cross product commutative?

No. It is anti-commutative: swapping the order flips the sign, so b⨯a = −(a⨯b).