Matrix Cross Product Calculator

Cross Product Formula:

\[ \vec{CP} = \det\begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{bmatrix} \]

Vector B (x):

Vector B (y):

Vector B (z):

Vector C (x):

Vector C (y):

Vector C (z):

Cross Product (CP):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Matrix Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space that results in a vector perpendicular to both original vectors. It's calculated using the determinant of a matrix composed of the unit vectors and the components of the two input vectors.

2. How Does the Calculator Work?

The calculator uses the matrix determinant formula:

\[ \vec{CP} = \det\begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{bmatrix} \]

Which expands to: \[ \vec{CP} = (B_yC_z - B_zC_y)\mathbf{i} - (B_xC_z - B_zC_x)\mathbf{j} + (B_xC_y - B_yC_x)\mathbf{k} \]

Explanation: The cross product produces a vector that is perpendicular to both input vectors, with magnitude equal to the area of the parallelogram they span.

3. Applications of Cross Product

Details: The cross product is essential in physics for calculating torque, in computer graphics for surface normals, and in engineering for determining moments of force. It's also used to find the normal vector to a plane defined by two vectors.

4. Using the Calculator

Tips: Enter the x, y, and z components for both vectors B and C. The calculator will compute the cross product vector CP = B × C.

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of the cross product?
A: The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, and its direction is perpendicular to both vectors according to the right-hand rule.

Q2: Is the cross product commutative?
A: No, the cross product is anti-commutative: A × B = - (B × A).

Q3: What happens when vectors are parallel?
A: If two vectors are parallel, their cross product is the zero vector (0, 0, 0).

Q4: Can cross product be calculated in 2D?
A: The cross product is specifically defined for 3D space. In 2D, we typically calculate the scalar value representing the "signed area" of the parallelogram.

Q5: What's the relationship between dot product and cross product?
A: While both involve two vectors, the dot product returns a scalar value, while the cross product returns a vector. They serve different mathematical purposes.