Cross Product Calculator

How to Calculate the Cross Product of Vectors

The cross product is a fundamental operation in vector algebra with applications in physics, engineering, and computer graphics. Here's a comprehensive guide on how to calculate the cross product of vectors in both 2D and 3D:

2D Cross Product

For two 2D vectors \(\vec{a} = (a_x, a_y)\) and \(\vec{b} = (b_x, b_y)\), their cross product is defined as:

\[ \vec{a} \times \vec{b} = a_x b_y - a_y b_x \]

The result is a scalar value representing the signed area of the parallelogram formed by the two vectors.

3D Cross Product

For two 3D vectors \(\vec{a} = (a_x, a_y, a_z)\) and \(\vec{b} = (b_x, b_y, b_z)\), their cross product is defined as:

\[ \vec{a} \times \vec{b} = (a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x) \]

The result is a vector perpendicular to both input vectors.

Calculation Steps

For 2D Vectors:

1. Identify the components of both vectors
2. Apply the formula: \(a_x b_y - a_y b_x\)
3. Interpret the scalar result

For 3D Vectors:

1. Identify the components of both vectors
2. Calculate the x-component: \(a_y b_z - a_z b_y\)
3. Calculate the y-component: \(a_z b_x - a_x b_z\)
4. Calculate the z-component: \(a_x b_y - a_y b_x\)
5. Combine the results into a new vector

Example Calculations

2D Example:

Let's calculate the cross product of two 2D vectors:

\(\vec{a} = (2, 3)\) and \(\vec{b} = (4, 5)\)

\[ \vec{a} \times \vec{b} = (2 \times 5) - (3 \times 4) = 10 - 12 = -2 \]

The cross product is -2, indicating the signed area of the parallelogram formed by these vectors.

3D Example:

Let's calculate the cross product of two 3D vectors:

\(\vec{a} = (2, 3, 4)\) and \(\vec{b} = (5, 6, 7)\)

1. X-component: \[ (3 \times 7) - (4 \times 6) = 21 - 24 = -3 \]
2. Y-component: \[ (4 \times 5) - (2 \times 7) = 20 - 14 = 6 \]
3. Z-component: \[ (2 \times 6) - (3 \times 5) = 12 - 15 = -3 \]
4. Combine the results: \[ \vec{a} \times \vec{b} = (-3, 6, -3) \]

Therefore, the cross product of \(\vec{a}\) and \(\vec{b}\) is \((-3, 6, -3)\).

Visual Representation

This diagram illustrates the cross product of 3D vectors \(\vec{a}\) and \(\vec{b}\). The resulting vector \(\vec{a} \times \vec{b}\) is perpendicular to both input vectors.