Vector Cross Calculator

Vector Cross Product Formula:

\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = \hat{i}(a_y b_z - a_z b_y) - \hat{j}(a_x b_z - a_z b_x) + \hat{k}(a_x b_y - a_y b_x) \]

Vector a (x-component):

Vector a (y-component):

Vector a (z-component):

Vector b (x-component):

Vector b (y-component):

Vector b (z-component):

Cross Product (a × b):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Vector Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space that results in another vector perpendicular to both original vectors. It has applications in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} \]

Which expands to:

\( \hat{i} \) component: \( a_y b_z - a_z b_y \)
\( \hat{j} \) component: \( -(a_x b_z - a_z b_x) \)
\( \hat{k} \) component: \( a_x b_y - a_y b_x \)

Explanation: The cross product magnitude equals the area of the parallelogram formed by the two vectors, and its direction follows the right-hand rule.

3. Applications of Cross Product

Details: Cross products are used to calculate torque, angular momentum, surface normals in 3D graphics, and electromagnetic fields in physics.

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The result will be a new vector perpendicular to both input vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar quantity while cross product gives a vector. Dot product measures parallelism, cross product measures perpendicularity.

Q2: Why is the cross product only defined in 3D?
A: The perpendicular vector concept only works consistently in three dimensions. In 2D, it gives a scalar, and in higher dimensions, more complex operations are needed.

Q3: What does a zero cross product mean?
A: A zero cross product indicates the vectors are parallel (or at least one is zero).

Q4: How is cross product related to right-hand rule?
A: The right-hand rule determines the direction of the resulting vector: point fingers in direction of first vector, curl toward second vector, thumb points in cross product direction.

Q5: Can cross product be commutative?
A: No, \( \vec{a} \times \vec{b} = -(\vec{b} \times \vec{a}) \). It's anti-commutative.