Cross Product Calculator

Cross Product Calculator

A Cross Product Calculator is a vital tool for students, engineers, and physicists working with 3D vectors in fields like mathematics, physics, and computer graphics. It computes the cross product of two vectors, such as a = [a₁, a₂, a₃] and b = [b₁, b₂, b₃], producing a vector perpendicular to both, essential for calculating torque, angular momentum, or 3D surface normals. Users input vector components, and the calculator delivers instant results.

The formula is: a × b = [(a₂b₃ – a₃b₂), (a₃b₁ – a₁b₃), (a₁b₂ – a₂b₁)]. For example, vectors a = [3, 1, 2] and b = [4, 5, 6] yield a × b = [(1×6 – 2×5), (2×4 – 3×6), (3×5 – 1×4)] = [-4, -10, 11]. The magnitude, |a × b| = |a||b|sin(θ), gives the area of the parallelogram formed, useful in mechanics or electromagnetism.

Benefits include error-free calculations, time savings, and clarity for complex applications like magnetic force or 3D rotations.

For accuracy, ensure correct component inputs and consistent units (e.g., meters for torque). Test scenarios with different vectors to verify orthogonality or explore properties. Limitations include reliance on numeric 3D vectors; non-numeric or 2D inputs may cause errors. A Cross Product Calculator simplifies vector analysis, enhancing efficiency in academic and professional tasks. Calculate your cross product now to streamline your work!

The Cross Product Calculator helps you compute the cross product (also known as the vector product) of two three-dimensional vectors. This mathematical operation results in a vector that is perpendicular to both of the original vectors. It’s commonly used in physics, engineering, and computer graphics to find normal, torque, and rotational effects.

What is a Cross Product?

In mathematics, the cross product of two vectors A and B (written as A × B) is a vector that is perpendicular to both A and B and has a magnitude equal to the area of the parallelogram that the vectors span.

It is only defined in three-dimensional space (3D).

Cross Product Formula

Let:

• A = (a₁, a₂, a₃)

• B = (b₁, b₂, b₃)

Then:

A × B = (a₂b₃ – a₃b₂, a₃b₁ – a₁b₃, a₁b₂ – a₂b₁)

The result is a new vector:
(x, y, z) which is perpendicular to both A and B.

A Cross Product Calculator is an essential tool for students, engineers, and physicists, computing the cross product of two 3D vectors in mathematics and physics. By inputting the components of two vectors (e.g., a = [a₁, a₂, a₃] and b = [b₁, b₂, b₃]), it calculates the resulting vector perpendicular to both, crucial for applications like torque, angular momentum, or surface normals in 3D graphics.

How Does a Cross Product Calculator Work?

The calculator uses the formula: a × b = [(a₂b₃ – a₃b₂), (a₃b₁ – a₁b₃), (a₁b₂ – a₂b₁)]. For example, for vectors a = [2, 3, 4] and b = [5, 6, 7], the cross product is [(3×7 – 4×6), (4×5 – 2×7), (2×6 – 3×5)] = [(21 – 24), (20 – 14), (12 – 15)] = [-3, 6, -3]. The result’s magnitude, computed as |a × b| = |a||b|sin(θ), represents the area of the parallelogram formed by the vectors.

Example

Let’s calculate:

A = (2, 3, 4)
B = (5, 6, 7)

Then:

A × B = (3×7 – 4×6, 4×5 – 2×7, 2×6 – 3×5)
    = (21 – 24, 20 – 14, 12 – 15)
    = (-3, 6, -3)

So, the cross product is (-3, 6, -3).