Cross Product Calculator

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Cross Product Calculator
Vector A × Vector B (Cross Product)
Vector A (u)
Vector B (v)
Result:
Enter vector components and click Calculate.
function calculateCrossProduct(){var ax=parseFloat(document.getElementById('ax').value);var ay=parseFloat(document.getElementById('ay').value);var az=parseFloat(document.getElementById('az').value);var bx=parseFloat(document.getElementById('bx').value);var by=parseFloat(document.getElementById('by').value);var bz=parseFloat(document.getElementById('bz').value);var showSteps=document.getElementById('steps').checked;if(isNaN(ax)||isNaN(ay)||isNaN(az)||isNaN(bx)||isNaN(by)||isNaN(bz)){document.getElementById('answer').innerHTML='Please enter valid numeric values for all vector components.';return;}var cx = (ay * bz) – (az * by);var cy = (az * bx) – (ax * bz);var cz = (ax * by) – (ay * bx);var magnitude = Math.sqrt(cx*cx + cy*cy + cz*cz);var output='
Vector A × Vector B = ('+cx+', '+cy+', '+cz+')
';output+='
Magnitude: '+magnitude.toFixed(4)+'
';if(showSteps){output+='
';output+='Steps:
';output+='i-component: (ay * bz) – (az * by) = ('+ay+' * '+bz+') – ('+az+' * '+by+') = '+cx+'
';output+='j-component: (az * bx) – (ax * bz) = ('+az+' * '+bx+') – ('+ax+' * '+bz+') = '+cy+'
';output+='k-component: (ax * by) – (ay * bx) = ('+ax+' * '+by+') – ('+ay+' * '+bx+') = '+cz+'
';output+='
';}document.getElementById('answer').innerHTML=output;}

Calculator Use

The cross product calculator is a specialized tool designed to find the vector product of two three-dimensional vectors. Unlike the dot product, which results in a scalar (a single number), the cross product results in a new vector that is perpendicular to both of the original vectors. This tool is essential for students, engineers, and physicists working with spatial geometry, torque, and electromagnetism.

To use this calculator, simply enter the x, y, and z components for Vector A and Vector B. The tool will automatically compute the resulting vector and its magnitude, with optional step-by-step breakdowns for educational purposes.

Vector A (u₁, u₂, u₃)
The first 3D vector, often represented in unit vector notation as u₁i + u₂j + u₃k.
Vector B (v₁, v₂, v₃)
The second 3D vector, which together with Vector A defines a plane in 3D space.
Result Vector (w₁, w₂, w₃)
The output vector, which is orthogonal (at a 90-degree angle) to both A and B.

How It Works

The cross product of two vectors A and B is calculated using the determinant of a 3×3 matrix where the first row contains the unit vectors i, j, and k. The mathematical formula for the cross product C = A × B is:

Cx = (Ay * Bz) – (Az * By)
Cy = (Az * Bx) – (Ax * Bz)
Cz = (Ax * By) – (Ay * Bx)

The properties of the result include:

  • Direction: Determined by the "Right-Hand Rule."
  • Magnitude: Equivalent to the area of the parallelogram formed by the two vectors.
  • Orthogonality: The result is always perpendicular to the plane containing the original two vectors.

Cross Product Example

Example: Find the cross product of Vector A = (2, 3, 4) and Vector B = (5, 6, 7).

Step-by-step solution:

  1. Identify components: Ax=2, Ay=3, Az=4; Bx=5, By=6, Bz=7.
  2. Calculate Cx: (3 * 7) – (4 * 6) = 21 – 24 = -3
  3. Calculate Cy: (4 * 5) – (2 * 7) = 20 – 14 = 6
  4. Calculate Cz: (2 * 6) – (3 * 5) = 12 – 15 = -3
  5. Final Result: (-3, 6, -3)

Common Questions

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

The dot product results in a scalar (a single number) and measures how much one vector "aligns" with another. The cross product results in a vector and measures the perpendicularity and the area formed by the vectors.

Can I calculate a cross product for 2D vectors?

Technically, the cross product is only defined for 3D vectors. However, you can calculate the "cross product" of two 2D vectors by treating them as 3D vectors with a Z-component of zero. The result will always point along the Z-axis.

Why is the order of vectors important?

The cross product is anticommutative, meaning A × B = -(B × A). Switching the order of the inputs will flip the direction of the resulting vector to the exact opposite direction.

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