# Why Is the Slope Between Perpendicular Lines -1, But Not for the X and Y Axes?

• Anukriti C.
In summary, the slope between two perpendicular lines on the Cartesian plane is -1, but this does not apply to the x and y axes. The y-axis has an undefined gradient, while the x-axis has a gradient of 0. When we multiply these two values together, we get an undefined result. However, the limit of two perpendicular lines approaching the axes is -1, but the actual value is undefined. This is similar to the equation y=x/x, where the limit is 1 but there is a hole in the graph at (0,1). It is important to note that the concept of "slope between two lines" is incorrect, and it should be referred to as "product of the slopes of two perpendicular lines".
Anukriti C.
we know that in the cartesian plane, slope between two perpendicular lines is -1. but what about the x and y axis? if we find the slope between them it is not equal to -1. why is the slope between two perpendicular lines on the cartesian plane is -1 but the axes themselves do not behave such?

Slope between ? How do you calculate that ?

Anukriti C. said:
we know that in the cartesian plane, slope between two perpendicular lines is -1. but what about the x and y axis? if we find the slope between them it is not equal to -1. why is the slope between two perpendicular lines on the cartesian plane is -1 but the axes themselves do not behave such?

Because the y-axis has an undefined gradient.

The slope of the x-axis has gradient 0, while the y-axis has an undefined gradient. Multiplying these two together gives us an undefined value, which basically says it's meaningless. It turns out in this particular case that the limit of two perpendicular lines that approach the axes will be -1, up until the point where they are the axes, at which the value becomes undefined.

Don't confuse these two though. The limit is -1, but the actual value is undefined.

This isn't any different to the equation y=x/x being equivalent to y=1 except at x=0 where it is undefined. The value of y in the limit as x approaches 0 is 1, but there is a hole in the graph at the point (0,1).

aikismos
Mentallic said:
Because the y-axis has an undefined gradient.

The slope of the x-axis has gradient 0, while the y-axis has an undefined gradient. Multiplying these two together gives us an undefined value, which basically says it's meaningless. It turns out in this particular case that the limit of two perpendicular lines that approach the axes will be -1, up until the point where they are the axes, at which the value becomes undefined.

Don't confuse these two though. The limit is -1, but the actual value is undefined.

This isn't any different to the equation y=x/x being equivalent to y=1 except at x=0 where it is undefined. The value of y in the limit as x approaches 0 is 1, but there is a hole in the graph at the point (0,1).
please explain what is meant by hole in the graph?
I'm really sorry but I didn't get what you are trying to say or maybe you didn't get what I am trying to ask.

BvU said:
Slope between ? How do you calculate that ?
we write the equations of the lines which are of the form y=mx+c where m in the slope or inclination of the line with x-axis.
we can also find it if we know a point lying on the line say(x,y) and the slope is m=tan theta= y/x
that's it!

Anukriti C. said:
we write the equations of the lines which are of the form y=mx+c where m in the slope or inclination of the line with x-axis.
we can also find it if we know a point lying on the line say(x,y) and the slope is m=tan theta= y/x
that's it!
So then you have the slope m1 of one line. And m2 of the other. What you mean to say is that if the lines are perpendicular, then m1 = -1/ m2 .

Which doesn't work if m1 = 0 or m2 = 0

The expression
Anukriti C. said:
slope between two perpendicular lines is -1
sort of sounded strange to me.

Anukriti C.
BvU said:
The expression
Anukriti C. said:
slope between two perpendicular lines is -1
sort of sounded strange to me.
And to me as well. You can talk about the angle between two perpendicular lines (which is 90°), but not the "slope" between them

Anukriti C.
Mark44 said:
And to me as well. You can talk about the angle between two perpendicular lines (which is 90°), but not the "slope" between them
but that is what we are taught. that's actually our common mathematics language and we talk about the slope between the lines. it's in books too.

No, that is NOT "what we are taught"! "slope" is a property of a single line, not "between" two lines. Your statement that "the slope between two lines is -1" is simply false and I certainly have never seen it in a textbook! What is true and is in any textbook I have ever seen is that "the product of the slopes of two perpendicular lines is -1", of course with the proviso that both of the lines have to have a slope so neither is vertical.

HallsofIvy said:
No, that is NOT "what we are taught"! "slope" is a property of a single line, not "between" two lines. Your statement that "the slope between two lines is -1" is simply false and I certainly have never seen it in a textbook! What is true and is in any textbook I have ever seen is that "the product of the slopes of two perpendicular lines is -1", of course with the proviso that both of the lines have to have a slope so neither is vertical.
I believe that was a bit harsh way of correction.
moreover, thanks for correcting me.
However, I think you have understood my original doubt and I wish if you could help me out
well, thanks for correcting me.
however, i wish if you could clear my original doubt which you have corrected, and a simple explanation would surely help me a lot.
P.S. I am not a very bright kid. please provide the explanation appropriately.

Last edited by a moderator:
Anukriti C. said:
I believe that was a bit harsh way of correction.
moreover, thanks for correcting me.
However, I think you have understood my original doubt and I wish if you could help me out
well, thanks for correcting me.
however, i wish if you could clear my original doubt which you have corrected, and a simple explanation would surely help me a lot.
P.S. I am not a very bright kid. please provide the explanation appropriately.
Your question has pretty much been answered, but I'll sum things up.
Anukriti C. said:
we know that in the cartesian plane, slope between two perpendicular lines is -1. but what about the x and y axis? if we find the slope between them it is not equal to -1. why is the slope between two perpendicular lines on the cartesian plane is -1 but the axes themselves do not behave such?
If L1 and L2 are two perpendicular lines in the plane, and neither is vertical, then the product of their slopes is -1. This is not the same as saying the "slope between the two lines is -1". In more detail, if the equation of L1 is y = m1x + b1, and the equation of L2 is y = m2x + b2, then ##m_1 m_2 = -1##. Equivalently, ##m_1 = -\frac 1 {m_2}##.

We don't allow either line to be vertical, because vertical lines have a slopt that is undefined. Also, the equation of every vertical line is x = k. With regard to the two coordinate axes in the plane, the x-axis is horizontal: its slope is 0. the y-axis is vertical: its slope is undefined. The equation of the x-axis is y = 0. You could also write this as y = 0x + 0, which emphasizes the fact that the slope is 0. The equation of the y-axis is x = 0. Since the slope is undefined, the equation of the y-axis cannot be put into the form y = mx + b.

Anukriti C.
Mark44 said:
Your question has pretty much been answered, but I'll sum things up.
If L1 and L2 are two perpendicular lines in the plane, and neither is vertical, then the product of their slopes is -1. This is not the same as saying the "slope between the two lines is -1". In more detail, if the equation of L1 is y = m1x + b1, and the equation of L2 is y = m2x + b2, then ##m_1 m_2 = -1##. Equivalently, ##m_1 = -\frac 1 {m_2}##.

We don't allow either line to be vertical, because vertical lines have a slopt that is undefined. Also, the equation of every vertical line is x = k. With regard to the two coordinate axes in the plane, the x-axis is horizontal: its slope is 0. the y-axis is vertical: its slope is undefined. The equation of the x-axis is y = 0. You could also write this as y = 0x + 0, which emphasizes the fact that the slope is 0. The equation of the y-axis is x = 0. Since the slope is undefined, the equation of the y-axis cannot be put into the form y = mx + b.
I'm going to rock the class now!(not exactly)
thanks a lot again

## 1. What is the formula for finding the slope between two points on a coordinate axis?

The formula for finding the slope (m) between two points (x1, y1) and (x2, y2) on a coordinate axis is:
m = (y2 - y1) / (x2 - x1).

## 2. How do I interpret the slope between two coordinate axes?

The slope between two coordinate axes represents the rate of change (or steepness) between the two points. A positive slope indicates an upward trend, while a negative slope indicates a downward trend.

## 3. Can the slope between two coordinate axes be greater than 1?

Yes, the slope between two coordinate axes can be greater than 1. This indicates a steep upward trend. For example, a slope of 2 means that for every 1 unit increase in the x-axis, there is a 2 unit increase in the y-axis.

## 4. Is a slope of 0 possible between two coordinate axes?

Yes, a slope of 0 is possible between two coordinate axes. This indicates a horizontal line, where there is no change in the y-axis for any change in the x-axis.

## 5. How can I use the slope between two coordinate axes in real-life applications?

The slope between two coordinate axes is commonly used in various fields such as physics, engineering, and economics. It can be used to calculate velocity, acceleration, and growth rates. For example, in economics, the slope can represent the marginal cost or revenue of a product. In physics, it can represent the speed or acceleration of an object.

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