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For \(\mathbf{a} = \langle a_1, a_2, a_3\rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3\rangle\), the cross product is \(\mathbf{a}\times\mathbf{b} = \langle a_2b_3 – a_3b_2,\ a_3b_1 – a_1b_3,\ a_1b_2 – a_2b_1\rangle\). It is a vector perpendicular to both. For example, \(\langle 1,2,3\rangle \times \langle 4,5,6\rangle = \langle -3, 6, -3\rangle\).
How to Use the Cross Product Calculator
- Type vector \(\mathbf{a}\) as three numbers, like
1, 2, 3. - Type vector \(\mathbf{b}\) in the second box.
- Press Solve to see the determinant setup, each component, the magnitude, and a check.
The Determinant Method
Write the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) in the top row, then the two vectors:
\[\mathbf{a}\times\mathbf{b} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\end{vmatrix}.\]
Expand along the top row. Remember the minus sign on the middle (\(\mathbf{j}\)) term.
Worked Example: \(\langle 1, 2, 3\rangle \times \langle 4, 5, 6\rangle\)
- \(\mathbf{i}\) component: \(2\cdot6 – 3\cdot5 = 12 – 15 = -3\)
- \(\mathbf{j}\) component: \(-(1\cdot6 – 3\cdot4) = -(6 – 12) = 6\)
- \(\mathbf{k}\) component: \(1\cdot5 – 2\cdot4 = 5 – 8 = -3\)
Answer: \(\langle -3, 6, -3\rangle\).
Check: \(\langle -3, 6, -3\rangle\cdot\langle 1, 2, 3\rangle = -3 + 12 – 9 = 0\) and \(\langle -3, 6, -3\rangle\cdot\langle 4, 5, 6\rangle = -12 + 30 – 18 = 0\). Both dot products are 0, so the result is perpendicular to both vectors. ✓
Properties of the Cross Product
| Property | Formula |
|---|---|
| Order matters | \(\mathbf{b}\times\mathbf{a} = -(\mathbf{a}\times\mathbf{b})\) |
| Parallel vectors | \(\mathbf{a}\times\mathbf{b} = \mathbf{0}\) |
| Magnitude | \(\|\mathbf{a}\times\mathbf{b}\| = \|\mathbf{a}\|\|\mathbf{b}\|\sin\theta\) |
| Area of a parallelogram | \(\|\mathbf{a}\times\mathbf{b}\|\) |
| Area of a triangle | \(\frac12\|\mathbf{a}\times\mathbf{b}\|\) |
| Unit vectors | \(\mathbf{i}\times\mathbf{j} = \mathbf{k}\), \(\mathbf{j}\times\mathbf{k} = \mathbf{i}\), \(\mathbf{k}\times\mathbf{i} = \mathbf{j}\) |
In the example, \(\|\langle -3, 6, -3\rangle\| = \sqrt{54} \approx 7.35\), so the parallelogram made by the two vectors has an area of about 7.35 square units.
The Right-Hand Rule
Point the fingers of your right hand along \(\mathbf{a}\) and curl them toward \(\mathbf{b}\). Your thumb points in the direction of \(\mathbf{a}\times\mathbf{b}\).
Common Mistakes
- Forgetting the minus sign on the \(\mathbf{j}\) component.
- Switching the order. \(\mathbf{a}\times\mathbf{b}\) and \(\mathbf{b}\times\mathbf{a}\) point in opposite directions.
- Mixing up dot and cross products. The dot product is a number. The cross product is a vector.
Learn the Math Behind the Cross Product Calculator
The cross product gives a vector perpendicular to two others. Its length is the area of the parallelogram they span.
- GuideHow to study calculus: a roadmap with checkpoints
- Key termsCurl, Determinant, Matrix
- Next toolsDeterminant Calculator, Curl Calculator, Matrix Calculator
Frequently Asked Questions
Can you take a cross product of 2D vectors?
Write them as 3D vectors with a 0 for the last component, like \(\langle 3, 4, 0\rangle\). The result points straight along the \(z\)-axis.
What is the cross product used for?
It finds a vector perpendicular to a plane, the area of a parallelogram or triangle, torque in physics, and normal vectors in 3D graphics.
Why is the cross product of parallel vectors zero?
Because \(\sin 0° = 0\). Parallel vectors make no area, so the magnitude is 0.