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.
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 = m_{1} x + b_{1}

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 m_{1} + b_{1})
B(7 , 7 m_{1} + b_{1})

We now find the components of vector AB.
vector(AB) = > 7  6 , (7 m_{1} + b_{1})  (6 m_{1} + b_{1}) >
= > 1 , m_{1}>

Let y = m_{2} x + b_{2} 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 m_{2} + b_{2})
B(5 , 5 m_{2} + b_{2})
vector(CD) = > 5  4 , (5 m_{2} + b_{2})  (4 m_{2} + b_{2}) >
= > 1 , m_{2}>

If the two lines are perpendicular then the two vectors AB and CD are orthogonal and their scalar product must be zero. Hence
> 1 , m_{1}> . > 1 , m_{2}> = 0

Which gives
1 + m_{1} * m_{2} = 0

The above may be written as
m_{1} * m_{2} =  1
m_{1} =  1 / m_{2} , m_{1} is equal to the negative reciprocal of m_{2}
m_{2} =  1 / m_{1} , m_{2} is equal to the negative reciprocal of m_{1}

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 ,
Match Linear Equations to Graphs. Excellent interactive activity where linear equations are matched to graphs.
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Equations of Line Through Two Points And Parallel and Perpendicular.
 
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Updated: February 2015
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