Cross Product Calculator

About Cross Product Calculator

The cross product of two vectors produces a vector perpendicular to both. Its magnitude equals the area of the parallelogram they span.

How to use this calculator

1. Enter the three components of the first vector, \(a_1\), \(a_2\), and \(a_3\).
2. Enter the three components of the second vector, \(b_1\), \(b_2\), and \(b_3\).
3. Click calculate to compute the cross product \(\vec{a} \times \vec{b}\).
4. Read the resulting vector, which will have three components and be perpendicular to both original vectors.

The formula explained

The formula computes the cross product of two vectors in 3D using a determinant setup. The result is a vector whose direction is perpendicular to both input vectors, and whose size depends on the angle between them.

• \(a_1, a_2, a_3\) = the x, y, and z components of the first vector
• \(b_1, b_2, b_3\) = the x, y, and z components of the second vector
• \(\vec{a} \times \vec{b}\) = the cross product vector
• \(\hat{i}, \hat{j}, \hat{k}\) = unit vectors in the x, y, and z directions

Step by step method

1. Write the determinant form with \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) on the first row.
2. Expand the determinant to get \((a_2b_3-a_3b_2)\hat{i} - (a_1b_3-a_3b_1)\hat{j} + (a_1b_2-a_2b_1)\hat{k}\).
3. Combine the three components into one vector.

Worked example

Problem. Find \(\vec{a} \times \vec{b}\) for \(\vec{a} = \langle 2, 3, 4 \rangle\) and \(\vec{b} = \langle 1, 0, 5 \rangle\).

1. Use the component formula: \(\langle a_2b_3-a_3b_2,\; a_3b_1-a_1b_3,\; a_1b_2-a_2b_1 \rangle\).
2. Substitute the numbers: \(\langle 3\cdot 5 - 4\cdot 0,\; 4\cdot 1 - 2\cdot 5,\; 2\cdot 0 - 3\cdot 1 \rangle = \langle 15, -6, -3 \rangle\).
3. So the cross product is \(\langle 15, -6, -3 \rangle\).

Tips and common mistakes

• The cross product is not commutative, so \(\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})\).
• If the vectors are parallel, the cross product is the zero vector because there is no perpendicular direction with area.

Frequently asked questions