1. What is 2D Vector Cross Product?

The 2D cross product (also called the perpendicular dot product) is an operation that takes two vectors and returns a scalar value representing the signed area of the parallelogram formed by the 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 value: Vector B is counter-clockwise from vector A
• Negative value: Vector B is clockwise from vector A
• Zero: Vectors are parallel

3. Importance of Cross Product

Details: The 2D cross product is useful in computer graphics, physics, and engineering for determining orientation, calculating areas, and finding perpendicular vectors.

4. Using the Calculator

Tips: Enter the x and y components of both vectors. The calculator will compute the cross product which can be used to determine the relative orientation of the vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between 2D and 3D cross products?
A: In 2D, the cross product returns a scalar representing signed area. In 3D, it returns a vector perpendicular to both input vectors.

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

Q3: How is this related to the determinant?
A: The 2D cross product is equivalent to the determinant of a 2×2 matrix formed by the two vectors.

Q4: Can I use this for 3D vectors?
A: For 3D vectors, you would need to calculate all three components of the cross product vector (x, y, z).

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