Cross Product Of Matrices Calculator

Cross Product Formula:

\[ \vec{C} = \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k} \]

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 Result:

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1. What Is The 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 has important applications in physics, engineering, and computer graphics.

2. How Does The Calculator Work?

The calculator uses the standard cross product formula:

Where:

\( \vec{A} = (A_x, A_y, A_z) \) — First input vector
\( \vec{B} = (B_x, B_y, B_z) \) — Second input vector
\( \vec{C} = (C_x, C_y, C_z) \) — Resulting cross product vector
\( \hat{i}, \hat{j}, \hat{k} \) — Unit vectors along x, y, z axes

Explanation: The cross product produces a vector 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 used to calculate torque in physics, determine surface normals in computer graphics, find magnetic force in electromagnetism, and solve various engineering problems involving rotational forces.

4. Using The Calculator

Tips: Enter the x, y, and z components of both vectors A and B. The calculator will compute the cross product vector C = A × B, which will be perpendicular to both input vectors.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between cross product and dot product?
A: Cross product gives a vector result (perpendicular to both inputs), while dot product gives a scalar result (related to the cosine of the angle between vectors).

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

Q3: What does a zero cross product indicate?
A: A zero cross product indicates that the two vectors are parallel or anti-parallel (angle of 0° or 180° between them).

Q4: Can cross product be calculated for 2D vectors?
A: The cross product is defined for 3D vectors only. For 2D vectors, the result would be a vector along the z-axis.

Q5: What is the geometric interpretation of cross product magnitude?
A: The magnitude |A × B| equals the area of the parallelogram spanned by vectors A and B.