Cross Product of Vectors Calculator

Cross Product Formula:

\[ \vec{a} \times \vec{b} = \langle a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x \rangle \]

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 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 cross product formula:

\[ \vec{a} \times \vec{b} = \langle a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x \rangle \]

Where:

\( \vec{a} = \langle a_x, a_y, a_z \rangle \) — First vector
\( \vec{b} = \langle b_x, b_y, b_z \rangle \) — Second vector
\( \times \) — Cross product operator

Explanation: The cross product produces a vector perpendicular to both input vectors, with magnitude equal to the product of their magnitudes and the sine of the angle between them.

3. Importance of Cross Product

Details: The cross product is essential in physics (torque, angular momentum), engineering (moment of force), and computer graphics (surface normals). It helps determine perpendicular vectors and areas/volumes in 3D space.

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The calculator will compute the resulting vector components. Results are rounded to 4 decimal places.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between cross product and dot product?
A: Cross product gives a vector perpendicular to both inputs, while dot product gives a scalar representing their parallel component.

Q2: What does a zero cross product mean?
A: A zero cross product indicates the vectors are parallel (or at least one is zero).

Q3: Is the cross product commutative?
A: No, \( \vec{a} \times \vec{b} = -(\vec{b} \times \vec{a}) \) (anti-commutative).

Q4: Can you compute cross product in 2D?
A: In 2D, the cross product is a scalar (z-component of the 3D result with z=0).

Q5: What's the geometric interpretation?
A: The magnitude equals the area of the parallelogram formed by the two vectors.