1. What is a Normal Vector?

A normal vector to a plane is a vector that is perpendicular to that plane. Given three non-collinear points in 3D space, there exists exactly one plane that passes through all three points, and this calculator finds the normal vector to that plane.

2. How Does the Calculator Work?

The calculator uses the cross product formula:

Where:

• \( P_1, P_2, P_3 \) — Three points on the plane
• \( \times \) — Cross product operation
• \( \vec{n} \) — Resulting normal vector

Explanation: The calculator first creates two vectors in the plane by subtracting coordinates, then computes their cross product which yields a perpendicular normal vector.

3. Importance of Normal Vectors

Details: Normal vectors are essential in computer graphics, physics simulations, engineering, and mathematics. They are used for lighting calculations, collision detection, and defining plane equations.

4. Using the Calculator

Tips: Enter the x, y, z coordinates of three non-collinear points. The points must not all lie on the same straight line for the calculation to be valid.

5. Frequently Asked Questions (FAQ)

Q1: What if my points are collinear?
A: The calculator will return a zero vector [0, 0, 0] since collinear points don't define a unique plane.

Q2: Does the order of points matter?
A: The order affects the direction of the normal vector (right-hand rule), but both directions are valid normals.

Q3: How can I get a unit normal vector?
A: Divide each component by the vector's magnitude: \( \hat{n} = \frac{\vec{n}}{|\vec{n}|} \).

Q4: What's the relationship to the plane equation?
A: The normal vector [a,b,c] gives the coefficients in the plane equation ax + by + cz + d = 0.

Q5: Can I use this for 2D vectors?
A: In 2D, the "normal" is a single perpendicular vector, not calculated via cross product.