Lecture Notes on Cartesian Coordinates and Linear Equations

Introduction

• Objectives:
  • Understand Cartesian coordinates: points, lines, slope.
  • Calculate Δy and Δx.
  • Find distance between two points on a line.
  • Determine the slope of a line segment.
  • Find the fourth vertex of a parallelogram.
  • Derive the equation of a line.
  • Identify line intersections.
  • Calculate the distance from a point to a line.

Basics of Cartesian Coordinates

• A point in the xy-plane is defined by an ordered pair (x, y).
• Δx: Change in x-coordinate from P1 to P2.
• Δy: Change in y-coordinate from P1 to P2.
• Distance (d): Calculated using the Pythagorean theorem: ( d = \sqrt{(Δx)^2 + (Δy)^2} ).

Slope of a Line

• Denoted by m: ( m = \frac{Δy}{Δx} ).
  • "Rise over run" interpretation.
  • Undefined when Δx = 0 (vertical line).

Example Calculations

• From P1 to P2:
  • ( Δx = x_2 - x_1 )
  • ( Δy = y_2 - y_1 )
  • Distance: ( \sqrt{(Δx)^2 + (Δy)^2} )
  • Slope (m): ( \frac{Δy}{Δx} )
• From P2 to P1:
  • Similar calculations with swapped signs for Δx and Δy.

Properties of Straight Lines

• Defined by two points.
• Inclination (φ): Angle between positive x-axis and the line.
  • ( \text{Slope} = \tan(φ) )
  • Horizontal line: φ = 0, slope = 0.
  • Vertical line: φ = 90°, slope is undefined.

Parallel and Perpendicular Lines

• Parallel Lines:
  • Same slope (m).
  • Vertical lines must both have undefined slopes.
• Perpendicular Lines:
  • Slopes are negative reciprocals.
  • Vertical and horizontal lines are perpendicular.

Line Equations

• Point-Slope Form: ( y - y_1 = m(x - x_1) )
• Slope-Intercept Form: ( y = mx + b )
• Two-Point Formula: Derived from slopes via two points.
• Intercept Form: ( \frac{x}{a} + \frac{y}{b} = 1 )
• General Form: ( ax + by = c )

Solving Line Problems

• Example: Finding a line's equation, intersections, and distances.
  • Convert given line equations to slope-intercept form.
  • Find intersection points using simultaneous equations.
  • Distance between a point and a line using perpendicular point calculation.

Application: Heat Conversion Problem

• Relationship between heat (H) and temperature (T) is linear.
• Use slope (m) to express H in terms of T:
  • Slope: Change in heat per change in temperature.
  • Equation: ( H = -\frac{8}{3}(T - 10) + 2480 )
• Calculate H at a specific temperature (e.g., 35°C).

Conclusion

• Application of Cartesian coordinates and linear equations across different problem contexts.