We are asked to discuss the similarities and differences for determining whether two lines are parallel or perpendicular.
How to Determine if Two Lines Are Parallel
Non-vertical lines are parallel when they have the same slope and different y-intercepts. We need to verify by looking at their equations if the slopes are the same. Let's consider the lines l_1, l_2, l_3, and l_4.
l_1&: y=2x-1
l_2&: y=2x+3
l_2&: y=2.5x-1
l_2&: y=1.5x+3Let's now determine whether l_1 and l_2 are parallel, and whether l_3 and l_4 are parallel.
Line
Equation
Slope
Are They Parallel?
l_1
y= 2x-1
2
Yes
l_2
y= 2x+3
2
l_3
y= 2.5x-1
2.5
No
l_4
y= 1.5x+3
1.5
Extra
Graphs of the Equations
As we can see from the graphs of the equations, parallel lines have the same slope.
How to Determine if Two Lines Are Perpendicular
Non-vertical lines are perpendicular when their slopes are opposite reciprocals. In other words, the condition is that the product of their slopes equals - 1. Let's consider the lines l_5, l_6, l_7, and l_8.
l_5&: y=2x
l_6&: y=- 1/2x
l_7&: y=2x-1
l_8&: y=0.5x+2
Let's now determine whether l_5 and l_6 are perpendicular, and whether l_7 and l_8 are perpendicular.
Line
Equation
Slope
m_1 * m_2
Are They Perpendicular?
l_5
y=2x
2
2 ( - 1/2 ) = - 1
Yes
l_6
y=- 1/2x
- 1/2
l_7
y=2x-1
2
2 ( 1/2 )= 1
No
l_8
y=0.5x+2
1/2
Extra
Graphs of the Equations
As we can see from the graphs of the equations, the slopes of perpendicular lines are opposite reciprocals.
Conclusion
Considering the mentioned points, we can make a few conclusions about the processes.
Similarities
Differences
Both processes rely on the value of the slopes of the lines.
The parallel condition needs us to check if the slopes are the same.
The condition for perpendicular lines relies on the products of the slopes. If the product is - 1, then the lines are perpendicular.
