Slopes of Two Perpendicular Lines

Problem:
Prove that two perpendicular lines have slopes that are negative reciprocal of each other.

Solution to Problem:

• The figure below shows two perpendicular lines L1 and L2.
  
  

  .
  
  

• Let the equation of line L1 be
  
  y = m₁ x + b₁
  
  
• We now select two points A and B on line L1 whose x coordinates are 6 and 8 respectively and find their y coordinates.
  
  A(6 , 6 m₁ + b₁)
  
  B(7 , 7 m₁ + b₁)
  
  
• We now find the components of vector AB.
  
  vector(AB) = < 7 - 6 , (7 m₁ + b₁) - (6 m₁ + b₁) >
  
  = < 1 , m₁ >
  
  
• Let y = m₂ x + b₂ be the equation of line L2. We now select two points C and D on line L2 whose x coordinates are 4 and 5 respectively and find their y coordinates.
  
  A(4 , 4 m₂ + b₂)
  
  B(5 , 5 m₂ + b₂)
  
  vector(CD) = < 5 - 4 , (5 m₂ + b₂) - (4 m₂ + b₂) >
  
  = < 1 , m₂ >
  
  
• If the two lines are perpendicular then the two vectors AB and CD are orthogonal and their scalar product must be zero. Hence
  
  < 1 , m₁ > . < 1 , m₂ > = 0
  
  
• Which gives
  
  1 + m₁ * m₂ = 0
  
  
• The above may be written as
  
  m₁ * m₂ = - 1
  
  m₁ = - 1 / m₂ , m₁ is equal to the negative reciprocal of m₂
  
  m₂ = - 1 / m₁ , m₂ is equal to the negative reciprocal of m₁
  
  
• NOTE: Points A, B, C and D and their x coordinates can choosen anywhere on the two lines. As an exercise choose other values for the x coordinates of points A, B, C and D and redo the calculation above. You should be able to make the same conclusion concerning the slopes of two perpendicular lines.

More math problems with detailed solutions ,

Tutorial on Equation of Line

and Line Problems