Section 5.3
Parallel and Perpendicular Lines
Working in a coordinate system, two non-vertical lines are parallel if, and only if, they have the same slope. Since vertical lines don't have such a thing as slope, they are excluded from this definition.
Two lines, neither of which is vertical, are perpendicular to each other if, and only if, their slopes are the negative reciprocals of each other.
Examples:
a. The lines $y=2x-3$ and $y=2x+7$ are parallel to each other because they both have slope $m=2$.
b. To write an equation of the line that passes through $P(2,-5)$ and is parallel to the line $y=\dfrac{3}{4}(x-1)$, we note that the slope of the given line is $m=\dfrac{3}{4}$. To write an equation of the line passing through $P(2,-5)$ with slope $m=\dfrac{3}{4}$, we use the Point-Slope Form of the equation of a line: $y+5=\dfrac{3}{4}(x-2)$.
c. The lines $3x+2y=6$ and $2x-3y=12$ are perpendicular to each other. The first equation can be rewritten $y=-\dfrac{3}{2}x+3$; the second equation can be rewritten $y=\dfrac{2}{3}x-4$. Since the slopes of these lines ($m=-\dfrac{3}{2}$ and $m=\dfrac{2}{3}$, respectively) are negative reciprocals of each other, the lines are perpendicular to each other.
d. Two vertical lines are parallel to each other. They will both have equations that can be put into the form $x=K$. For instance, the two lines $x=3$ and $x=-5$ are both vertical; they are parallel to each other.
e. Two lines, one horizontal and the other vertical, are perpendicular to each other. The horizontal line has an equation of the form $y=H$; it has a slope of 0. The vertical line has an equation of the form $x=K$; the slope of this line is undefined. The lines $x=3$ and $y=-2$ are perpendicular to each other.
Equations of parallel and perpendicular lines: Equations of Parallel and Perpendicular Lines
Exercises:
Write equations for each of the following lines.
1. Parallel to the line $y=3x-2$ and having y-intercept $5$.
2. Parallel to the line $y=-\tfrac{2}{3}x+4$ and having y-intercept $-2$.
3. Parallel to the line $y-2=\tfrac{1}{2}(x-4)$ and having y-intercept $1$.
4. Parallel to the line $y+3=-\tfrac{3}{5}(x+1)$ and having y-intercept $2$.
5. Parallel to the line $y+7=\tfrac{4}{7}(x-1)$ and having y-intercept $0$.
6. Parallel to the line $2x-3y=4$ and passing through $P(1,2)$.
7. Parallel to the line $3x+2y=8$ and passing through $P(-1,3)$.
8. Parallel to the line $x-3y=8$ and passing through $P(-11,4)$.
9. Parallel to the line $x+5y=1$ and passing through $P(0,4)$.
10. Parallel to the line $y=12$ and passing through $P(0,4)$.
11. Parallel to the line $x=-3$ and passing through $P(2,-2)$.
12. Parallel to the line $\frac{x}{3}-\frac{y}{5}=1$ and passing through $P(2,-2)$.
13. Parallel to the line $\frac{3x}{2}+\frac{2y}{3}=1$ and passing through $P(-1,-3)$.
14. Perpendicular to the line $y=3x-2$ and having y-intercept $4$.
15. Perpendicular to the line $y=\tfrac{2}{3}x-5$ and having y-intercept $3$.
16. Perpendicular to the line $y=-4x+1$ and having y-intercept $-1$.
17. Perpendicular to the line $2x+3y=12$ and passing through $P(-1,-3)$.
18. Perpendicular to the line $3x-4y=5$ and passing through $P(2,-2)$.
19. Perpendicular to the line $y-6=\tfrac{5}{3}(x+1)$ and passing through $P(-11,4)$.
20. Perpendicular to the line $y+1=-\tfrac{2}{3}(x-3)$ and having the same y-intercept as that line.
21. Perpendicular to the line $y=7$ and passing through $P(-1,-3)$.
22. Consider the two lines $3x-2y=7$ and $y=\tfrac{2}{3}x+5$. Are these lines parallel, perpendicular, or neither? Why?
23. Consider the two lines $4x-3y=12$ and $y-2=\tfrac{4}{3}(x+1)$. Are these lines parallel, perpendicular, or neither? Why?
24. Consider the two lines $2x+5y=10$ and $5x-2y=10$. Are these lines parallel, perpendicular, or neither? Why?
25. Consider the line $\ell $ graphed below. Write equations of each of the following.
a. The line through $P$ and parallel to $\ell $.
b. The line through $P$ and perpendicular to $\ell $.
c. The line through $Q$ and parallel to $\ell $.
d. The line through $Q$ and perpendicular to $\ell $.