Cross Product of 2 3D Vectors Calculator Desmos

3D Vector Cross Product Formula:

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

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 (A × B):

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1. What is the 3D Vector 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. Its magnitude relates to the area of the parallelogram spanned by the two vectors.

2. How Does the Calculator Work?

The calculator uses the determinant formula for cross product:

\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \\ \end{vmatrix} \]

Which expands to:

X-component: \( A_yB_z - A_zB_y \)
Y-component: \( -(A_xB_z - A_zB_x) \)
Z-component: \( A_xB_y - A_yB_x \)

3. Importance of Cross Product

Applications: The cross product is essential in physics (torque, angular momentum), computer graphics (surface normals), and engineering (moment of forces).

4. Using the Calculator

Instructions: Enter the x, y, z components for both vectors A and B. The calculator will compute the resulting cross product vector.

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 perpendicular to both input vectors.

Q2: What does the cross product magnitude represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q3: Why is the cross product only defined in 3D?
A: The perpendicular vector concept only works consistently in three dimensions, though 7D has a similar operation.

Q4: How is this related to Desmos?
A: This calculator helps verify hand calculations before implementing them in Desmos 3D graphing.

Q5: What's the right-hand rule?
A: Point fingers in direction of A, curl towards B; thumb points in direction of A × B.