Parallel and Perpendicular Lines
Parallel Lines (figure 1)
Parallel Lines (figure 1)
Perpendicular Lines (figure 2)
Perpendicular Lines (figure 2)
Before going through this page, you need to know what the equation of straight line is and how to find them. Check out the Straight Lines Part 1 page and Straight Lines Part 2 if you don't.
Parallel lines are two lines that never intersect (meet). As shown in figure 1, the lines y = 2x - 5 and y = 2x + 2 are parallel to each other, in other words they will NEVER INTERSECT.
We can show this algebraically as well.
If the line y = 2x - 5 and and the line y = 2x + 2 intersect, it means there is a value of x and a value y which satisfy both equations simultaneously. If there is a value for both x and y, it means there is an input value which when substituted with x in both equations leads to the same output (y) value. This would mean that the output would equal 2x - 5 AND 2x + 2, given this we can create a linear equation to find the x value.
2x - 5 = 2x + 2 (what we have written on the left here is wrong though, we should use the NOT EQUAL TO sign)
There is no solution to this equation, why?
If we multiply any number with 2 and then take away 5, how can it be the same as multiplying that same number by 2 and then adding 2. READ EACH SIDE CAREFULLY.
So we should now write that
2x - 5 ≠ 2x + 2
Given that there is no solution, it means that the lines do not intersect.
YOU MIGHT BE THINKING, why do we need to show algebraically that these two lines will never meet, we can just draw the lines and show that they will never intersect. But showing this algebraically tells us an important thing.
2x - 5 ≠ 2x + 2
We cannot solve for x here, as there is the same x (input) terms on both sides with different y-intercepts. If we cannot solve to get a value for x, we cannot get a value for y, and thus there is no solution for a value of x and a value of y, which simultaneously satisfies both equations. It means that there is no input value which can be substituted with x in both equations which gives the same output value of both equations. If there is no input and output value, it means there is no point/coordinate where the two lines intersect.
What causes the two lines to be parallel?
The two lines are parallel when the coefficient of the input is the same on both sides with different y-intercepts. In other words, the coefficient of the input is determined by the gradient of the line, so two lines are parallel when THE GRADIENTS OF THE LINES ARE THE SAME and they have different y-intercepts
Lets try another question
Are the lines y = -3x + 4 and y = -3x + 1 parallel. Yes they are, as both equations have the same gradient, m = -3 in both equations.
Can we prove algebraically, that there is no x and y value which simultaneously satisfies both equations? Yes we can.
-3x + 4 ≠ -3x + 1
We cannot solve this as both sides do not equal, as the coefficient of the input (x) is the same on both sides and each side has different y-intercepts. Thus if there is no value of x, we cannot solve to find the value of y. If there is no input and output value, there is no intersection point/coordinate of the lines, so the lines are parallel.
Perpendicular lines are two lines which when they intersect they form a 90 degrees angle, as shown in figure 2. The two lines shown in figure 2 are y = (-1/2)x + 3 and y = 2x - 5. How do we know if two lines are perpendicular? We can look at their gradients.
If the gradient of one line is equal to a (where a represents a number) and the gradient of another line is 1/-a, then these two lines are perpendicular.
In our case the gradient of y = 2x - 5 is 2, (here a = 2) so in order for a line to be perpendicular with this line, the gradient of that line HAS TO BE 1/-a which is equal to 1/-2 = -1/2.
y = (-1/2)x + 3 does have a gradient equal to -1/2, thus the lines are perpendicular.
Are the lines y = -5x + 3 and y = (1/5)x - 20 perpendicular?
Yes.
If the gradient of one line is equal to a, in our case a = 5, then for a line to be perpendicular with this line, the gradient of that line should be 1/-a. So what does 1/-a equal, when a = 5? I t equals 1/-5 = -1/5 (this is our gradient of the other line. So they are perpendicular.
Is the line y = (2/3)x -2 perpendicular to the line y = (2/3)x + 5?
By examining the gradients, we can see that they are the same. The equations have different y-intercepts and the same gradients. WHAT DOES THIS MEAN? IT MEANS THEY ARE PARALLEL NOT PERPENDICULAR.
For a line to be perpendicular to the line y = (2/3)x + 5, the gradient of that line needs to be -3/2. Why?
Remember if the gradient of line is equal to a, then for another line to be perpendicular, the gradient would have to equal 1/-a.
In our case a = 2/3, so 1/-a = 1/(-2/3) = 1 ÷ -2/3 = 1 x 3/-2 = 3/-2
Thus the gradient of a line which is perpendicular to the line y = (2/3)x + 5 needs to be 3/-2.
Parallel Lines
Parallel Lines
Parallel lines are two lines that never intersect (meet). As shown in figure 1, the lines y = 2x - 5 and y = 2x + 2 are parallel to each other, in other words they will NEVER INTERSECT.
We can show this algebraically as well.
If the line y = 2x - 5 and and the line y = 2x + 2 intersect, it means there is a value of x and a value y which satisfy both equations simultaneously. If there is a value for both x and y, it means there is an input value which when substituted with x in both equations leads to the same output (y) value. This would mean that the output would equal 2x - 5 AND 2x + 2, given this we can create a linear equation to find the x value.
2x - 5 = 2x + 2 (what we have written on the left here is wrong though, we should use the NOT EQUAL TO sign)
There is no solution to this equation, why?
If we multiply any number with 2 and then take away 5, how can it be the same as multiplying that same number by 2 and then adding 2. READ EACH SIDE CAREFULLY.
So we should now write that
2x - 5 ≠ 2x + 2
Given that there is no solution, it means that the lines do not intersect.
YOU MIGHT BE THINKING, why do we need to show algebraically that these two lines will never meet, we can just draw the lines and show that they will never intersect. But showing this algebraically tells us an important thing.
2x - 5 ≠ 2x + 2
We cannot solve for x here, as there is the same x (input) terms on both sides with different y-intercepts. If we cannot solve to get a value for x, we cannot get a value for y, and thus there is no solution for a value of x and a value of y, which simultaneously satisfies both equations. It means that there is no input value which can be substituted with x in both equations which gives the same output value of both equations. If there is no input and output value, it means there is no point/coordinate where the two lines intersect.
What causes the two lines to be parallel?
The two lines are parallel when the coefficient of the input is the same on both sides with different y-intercepts. In other words, the coefficient of the input is determined by the gradient of the line, so two lines are parallel when THE GRADIENTS OF THE LINES ARE THE SAME and they have different y-intercepts
Lets try another question
Are the lines y = -3x + 4 and y = -3x + 1 parallel. Yes they are, as both equations have the same gradient, m = -3 in both equations.
Can we prove algebraically, that there is no x and y value which simultaneously satisfies both equations? Yes we can.
-3x + 4 ≠ -3x + 1
We cannot solve this as both sides do not equal, as the coefficient of the input (x) is the same on both sides and each side has different y-intercepts. Thus if there is no value of x, we cannot solve to find the value of y. If there is no input and output value, there is no intersection point/coordinate of the lines, so the lines are parallel.
Perpendicular Lines
Perpendicular Lines
Perpendicular lines are two lines which when they intersect they form a 90 degrees angle, as shown in figure 2. The two lines shown in figure 2 are y = (-1/2)x + 3 and y = 2x - 5. How do we know if two lines are perpendicular? We can look at their gradients.
If the gradient of one line is equal to a (where a represents a number) and the gradient of another line is 1/-a, then these two lines are perpendicular.
In our case the gradient of y = 2x - 5 is 2, (here a = 2) so in order for a line to be perpendicular with this line, the gradient of that line HAS TO BE 1/-a which is equal to 1/-2 = -1/2.
y = (-1/2)x + 3 does have a gradient equal to -1/2, thus the lines are perpendicular.
Are the lines y = -5x + 3 and y = (1/5)x - 20 perpendicular?
Yes.
If the gradient of one line is equal to a, in our case a = 5, then for a line to be perpendicular with this line, the gradient of that line should be 1/-a. So what does 1/-a equal, when a = 5? I t equals 1/-5 = -1/5 (this is our gradient of the other line. So they are perpendicular.
Is the line y = (2/3)x -2 perpendicular to the line y = (2/3)x + 5?
By examining the gradients, we can see that they are the same. The equations have different y-intercepts and the same gradients. WHAT DOES THIS MEAN? IT MEANS THEY ARE PARALLEL NOT PERPENDICULAR.
For a line to be perpendicular to the line y = (2/3)x + 5, the gradient of that line needs to be -3/2. Why?
Remember if the gradient of line is equal to a, then for another line to be perpendicular, the gradient would have to equal 1/-a.
In our case a = 2/3, so 1/-a = 1/(-2/3) = 1 ÷ -2/3 = 1 x 3/-2 = 3/-2
Thus the gradient of a line which is perpendicular to the line y = (2/3)x + 5 needs to be 3/-2.