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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