Cross Product of Two Vectors - Calculator

An online calculator that calculates the cross product of two vectors, given by their components, is presented.

Definition of Cross Product of two Vectors

Let \(\mathbf{u}\) and \(\mathbf{v}\) be two vectors given by their 3 components as follows \[ \mathbf{u} = \langle a,\, b,\, c \rangle \quad \text{and} \quad \mathbf{v} = \langle d,\, e,\, f \rangle. \] The cross product of the two vectors \(\mathbf{u}\) and \(\mathbf{v}\), given above, is another vector \(\mathbf{w}\) given by \[ \mathbf{w} = \mathbf{u} \times \mathbf{v} = \langle a,\, b,\, c \rangle \times \langle d,\, e,\, f \rangle = \langle x,\, y,\, z \rangle, \] with the components \(x\), \(y\), and \(z\) of vector \(\mathbf{w}\) given by: \[ \begin{aligned} x &= b \times f - c \times e, \\ y &= c \times d - a \times f, \\ z &= a \times e - b \times d. \end{aligned} \]

Exanmple 1

Let \(\mathbf{u} = \langle 2,\, 2,\, 0 \rangle\) and \(\mathbf{v} = \langle 2,\, -2,\, 0 \rangle\). Then \(\mathbf{w} = \mathbf{u} \times \mathbf{v} = \langle x,\, y,\, z \rangle\), where \[ \begin{aligned} x &= 2 \times 0 - 0 \times (-2) = 0, \\ y &= 0 \times 2 - 2 \times 0 = 0, \\ z &= 2 \times (-2) - 2 \times 2 = -8. \end{aligned} \] Hence \[ \mathbf{w} = \langle 0,\, 0,\, -8 \rangle. \] Below are shown the plots of vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w} = \mathbf{u} \times \mathbf{v}\). Orthogonal 3 Dimensional Vectors

Use of Cross Product Calculator

1 - Enter the components of each of the two vectors \( \mathbf{u} \) and \( \mathbf{v} \) , as real numbers in decimal from and press "Calculate Cross Product". The answer is a vector \( \mathbf{u} \).
No characters other than real numbers are accepted by the calculator.

Applications of the Cross Product

The cross product of two vectors has applications in mathematics, physics, engineering, ... Some examples are presented below.

Example 2

One of the formulas to find the area of a triangle with sides having lengths \(|\mathbf{u}|\) and \(|\mathbf{v}|\) and an angle between them equal to \(\theta\) is \[ \text{Area}_{\triangle} = \frac{1}{2}\,|\mathbf{u}|\,|\mathbf{v}|\,\sin(\theta). \] Also, it can be shown that \[ |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}|\,|\mathbf{v}|\,\sin(\theta). \] We can therefore conclude that the area of a parallelogram spanned by vectors \(\mathbf{u}\) and \(\mathbf{v}\) is equal to the magnitude of the cross product of \(\mathbf{u}\) and \(\mathbf{v}\): \[ \text{Area}_{\text{parallelogram}} = |\mathbf{u} \times \mathbf{v}|. \] Cross Product and Area of Parallelogram

Example 3 - Application of Cross Product in Physics

When a particle of charge \(q\) is moving at a velocity \(\mathbf{v}\) in a magnetic field \(\mathbf{B}\), a force \(\mathbf{F}\) acts on this charge and is given by \[ \mathbf{F} = q\,(\mathbf{v} \times \mathbf{B}). \] Here, \(q\) is a scalar, while \(\mathbf{v}\) and \(\mathbf{B}\) are vectors; the expression \(\mathbf{v} \times \mathbf{B}\) denotes the cross product of \(\mathbf{v}\) and \(\mathbf{B}\).