Lecture Notes: Perpendicular Lines

Definition

• Perpendicular lines: Lines that intersect at a 90-degree angle.
• Example: Line y = 2x - 2 is perpendicular to line y = -1/2x + 8 as they intersect at a right angle.

Identifying Perpendicular Lines

• Visual Indicator: A right angle is often marked by a small square.
• Mathematical Verification: The product of the gradients (slopes) of two perpendicular lines should equal -1.
  • Formula: m1 * m2 = -1
  • m1: Gradient of the first line
  • m2: Gradient of the second line*

Example Calculation

1. Given Lines:
   • First line: Gradient m1 = 2
   • Second line: Gradient m2 = -1/2
2. Verification:
   • Calculation: 2 * -1/2 = -1
   • Conclusion: These lines are perpendicular.*

Practice Questions

• Question 1: Gradients are 1/2 and 3
  • Calculation: 1/2 * 3 = 3/2 (not -1)
  • Conclusion: Lines are not perpendicular.
• Question 2: Convert equation 2y = 3x + 8
  • Convert to y = 3/2x + 4, gradients are 3/2 and -2/3
  • Calculation: 3/2 * -2/3 = -6/6 = -1
  • Conclusion: Lines are perpendicular.

Finding Gradient of Perpendicular Line

• Given: Line y = 1/3x + 3
• To find the gradient of a perpendicular line:
  • Use formula: m2 = -1/m1
  • Given m1 = 1/3, calculate m2 = -1/(1/3) = -3

Finding the Equation of a Perpendicular Line

• Requirements:
  • Gradient of the line
  • Coordinates of a point through which it passes
• Note: Details on finding the equation are covered in another video.

Conclusion

• Understanding perpendicular lines involves both visual indicators and mathematical calculations.
• For more detailed calculations and the equation of a line, refer to additional resources.
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