Slopes of Two Perpendicular Lines
A detailed tutorial on how to prove that the slopes of two perpendicular lines are the negative reciprocal of each other.
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.
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Updated: 2 April 2013
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