1. What is an Orthogonal Trajectory?

An orthogonal trajectory is a family of curves that intersect another family of curves at right angles. In mathematics, they are important in fields like differential equations and physics.

2. How Does the Calculator Work?

The calculator uses the orthogonal trajectory principle:

Where:

• \( \frac{dy}{dx} \) — Slope of the original curve
• \( -\frac{1}{\frac{dy}{dx}} \) — Slope of the orthogonal trajectory

Explanation: The slope of an orthogonal trajectory is the negative reciprocal of the original slope.

3. Mathematical Principle

Details: For two curves to be orthogonal, the product of their slopes at the point of intersection must equal -1. This comes from the perpendicularity condition in analytic geometry.

4. Using the Calculator

Tips: Enter the slope of the original curve. This can be a numeric value (like 2 or 0.5), a fraction (like 3/4), or an algebraic expression (like x/y).

5. Frequently Asked Questions (FAQ)

Q1: What if the original slope is zero?
A: The orthogonal trajectory would be a vertical line (undefined slope).

Q2: What if the original slope is undefined (vertical line)?
A: The orthogonal trajectory would be a horizontal line (slope = 0).

Q3: Can I use this for implicit functions?
A: Yes, as long as you can determine the slope dy/dx at a point, you can find the orthogonal trajectory.

Q4: How is this used in real applications?
A: Orthogonal trajectories appear in physics (equipotential lines), engineering (heat flow), and mathematics (isogonal trajectories).

Q5: What about more complex curves?
A: For complex curves, you would first need to find dy/dx through implicit differentiation before applying this method.