Ratios Of Directed Line Segments Calculator With Angles

Directed Line Segments Formula:

\[ \text{Ratio} = \frac{AP}{PB} = \frac{m}{n} \] \[ \text{Point P} = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] \[ \text{Angle} = \arctan\left(\frac{y_2 - y_1}{x_2 - x_1}\right) \]

Point A X-coordinate:

Point A Y-coordinate:

Point B X-coordinate:

Point B Y-coordinate:

Ratio m:

Ratio n:

Point P Coordinates:

Ratio AP:PB:

Angle of Line AB:

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1. What Are Directed Line Segments?

A directed line segment is a line segment between two points with a specific direction (from point A to point B). The ratio of division determines where a point P divides the segment AB internally.

2. How Does the Calculator Work?

The calculator uses the section formula and angle calculation:

\[ \text{Point P} = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] \[ \text{Angle} = \arctan\left(\frac{y_2 - y_1}{x_2 - x_1}\right) \]

Where:

\( A(x_1,y_1) \) — Starting point coordinates
\( B(x_2,y_2) \) — Ending point coordinates
\( m:n \) — Ratio in which P divides AB
Angle — Angle of line AB with the positive x-axis

3. Importance of Ratio Calculation

Details: Calculating ratios of directed line segments is fundamental in geometry, physics, and computer graphics for interpolation, vector calculations, and position determination.

4. Using the Calculator

Tips: Enter coordinates of points A and B, and the ratio m:n. The calculator will determine the dividing point P and the angle of line AB.

5. Frequently Asked Questions (FAQ)

Q1: What does a ratio m:n mean?
A: It means point P divides AB such that AP/PB = m/n. For m=n=1, P is the midpoint.

Q2: How is the angle calculated?
A: The angle is calculated between the line AB and the positive x-axis, measured counterclockwise.

Q3: What if the ratio is negative?
A: Negative ratios indicate external division (point P lies outside AB). This calculator handles only positive ratios.

Q4: Can I use 3D coordinates?
A: This calculator works only for 2D coordinates. For 3D, you would need to add z-coordinates.

Q5: What's the range of the angle?
A: The angle ranges from -180° to 180°, with positive x-axis as 0°.