1. What is a 2D Vector Cross Product?

The 2D cross product (also called the perpendicular dot product) is a scalar value that represents the signed area of the parallelogram formed by two vectors in 2D space. It's useful for determining the relative orientation of vectors.

2. How Does the Calculator Work?

The calculator uses the 2D cross product formula:

Where:

• \( A_x \) — x-component of vector A
• \( A_y \) — y-component of vector A
• \( B_x \) — x-component of vector B
• \( B_y \) — y-component of vector B

Explanation: The result is a scalar value that indicates:

• Positive: Vector B is counter-clockwise from vector A
• Zero: Vectors are parallel
• Negative: Vector B is clockwise from vector A

3. Importance of Cross Product

Applications: The 2D cross product is used in computer graphics, physics simulations, computational geometry, and for determining the orientation of points relative to lines.

4. Using the Calculator

Tips: Enter the x and y components of both vectors. The calculator will compute the cross product, which represents the signed area of the parallelogram formed by the two vectors.

5. Frequently Asked Questions (FAQ)

Q1: How is 2D cross product different from 3D?
A: In 3D, the cross product returns a vector perpendicular to both input vectors. In 2D, it returns a scalar representing the signed area.

Q2: What does the sign of the result mean?
A: Positive indicates counter-clockwise orientation from A to B, negative indicates clockwise, and zero means the vectors are parallel.

Q3: Can I use this for 3D vectors?
A: No, this calculator is specifically for 2D vectors. For 3D vectors, you would need to compute the full vector cross product.

Q4: What's the relationship to the determinant?
A: The 2D cross product is equivalent to the determinant of a 2×2 matrix formed by the two vectors.

Q5: How is this used in computer graphics?
A: It's commonly used for back-face culling, collision detection, and determining point-line relationships.