Cross Product Calculator Vectors

Cross Product Formula:

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

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 (A × B):

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1. What is the Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both original vectors, with magnitude equal to the area of the parallelogram that the vectors span.

2. How Does the Calculator Work?

The calculator uses the determinant formula for cross product:

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

Which expands to:

\( x \)-component: \( A_yB_z - A_zB_y \)
\( y \)-component: \( A_zB_x - A_xB_z \)
\( z \)-component: \( A_xB_y - A_yB_x \)

3. Importance of Cross Product

Applications: The cross product is essential in physics (torque, angular momentum), engineering (moment of force), computer graphics (surface normals), and many other fields involving 3D geometry.

4. Using the Calculator

Instructions: Enter all six components (x, y, z for both vectors A and B). The calculator will compute the resulting vector that is 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 and measures projection, while cross product gives a vector quantity and measures perpendicularity.

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

Q3: What's the right-hand rule?
A: Point fingers in direction of first vector, curl towards second vector; thumb points in cross product direction.

Q4: Can you compute cross product in 2D?
A: Technically no, but you can treat 2D vectors as having z=0 to get a result with only z-component.

Q5: What if the cross product is zero?
A: A zero cross product means the vectors are parallel (or at least one is zero).