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

[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?

 

[BvU]

Slope between ? How do you calculate that ?

 

[Mentallic]

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).

 

[Anukriti C.]

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.
however, I appreciate your effort.

 

[Anukriti C.]

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!

 

[BvU]

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

slope between two perpendicular lines is -1

sort of sounded strange to me.

 

[Mark44]

The expression

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.]

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.

 

[HallsofIvy]

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.

 

[Anukriti C.]

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.

 

[Mark44]

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.

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.]

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.

that was very much helpful.
I'm going to rock the class now!(not exactly)
thanks a lot again