1. What is the Normal Line Equation?

The normal line to a curve at a given point is the line perpendicular to the tangent line at that point. The equation is derived from the point-slope form using the negative reciprocal of the tangent's slope.

2. How Does the Calculator Work?

The calculator uses the normal line equation:

Where:

• \( (x_0, y_0) \) — Point on the curve
• \( m \) — Slope of the tangent line at that point
• \( -\frac{1}{m} \) — Slope of the normal line (negative reciprocal)

Explanation: The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal of the tangent's slope.

3. Importance of Normal Line Calculation

Details: Normal lines are important in physics (for calculating reflection angles), engineering (for stress analysis), and computer graphics (for lighting calculations).

4. Using the Calculator

Tips: Enter the point coordinates (x₀, y₀) and the slope of the tangent line (m). The slope cannot be zero (as the normal would be vertical with undefined slope).

5. Frequently Asked Questions (FAQ)

Q1: What if the tangent line is horizontal?
A: A horizontal tangent (m=0) would make the normal line vertical (undefined slope), which this calculator cannot handle.

Q2: What if the tangent line is vertical?
A: A vertical tangent has undefined slope, so its normal would be horizontal (slope=0).

Q3: How is this different from the tangent line equation?
A: The tangent line uses slope m, while the normal line uses slope -1/m.

Q4: Can I use this for 3D surfaces?
A: No, this is for 2D curves. 3D surfaces require calculating normal vectors.

Q5: What's the relationship with orthogonal trajectories?
A: Orthogonal trajectories are curves that intersect another family of curves at right angles, using the same perpendicularity principle.