1. What is the Normal Line Formula?

The normal line formula describes a line that is perpendicular to a tangent line at a given point on a curve. It's represented parametrically as r(t) = P + t·n, where P is a point on the line, n is the normal vector, and t is a parameter.

2. How Does the Calculator Work?

The calculator uses the normal line equation:

Where:

• \( r(t) \) — Position vector of the line
• \( P \) — Point on the line (P_x, P_y)
• \( n \) — Normal vector (n_x, n_y)
• \( t \) — Parameter value

Explanation: The equation generates points along the normal line by scaling the normal vector n with parameter t and adding it to point P.

3. Applications of Normal Lines

Details: Normal lines are essential in computer graphics, physics (for calculating reflections), engineering (stress analysis), and optimization problems (gradient descent).

4. Using the Calculator

Tips: Enter the point coordinates, normal vector components, and a t value to calculate a specific point on the normal line. The normal vector determines the direction of the line.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between normal and tangent lines?
A: A tangent line runs parallel to the curve at a point, while a normal line is perpendicular to the tangent at that point.

Q2: How do I find the normal vector?
A: For a curve y=f(x), the normal vector at point (a,b) is (-f'(a), 1) or (f'(a), -1).

Q3: Can this be used in 3D space?
A: The concept extends to 3D, but this calculator is for 2D. The 3D version would need z-components.

Q4: What does the parameter t represent?
A: t is a scalar that determines how far along the normal line you move from point P.

Q5: How is this related to gradient vectors?
A: For level curves, the gradient vector is normal to the curve, so it can be used as the normal vector n.