The discussion revolves around the concept of slope in the Cartesian plane, specifically addressing why the slope between two perpendicular lines is -1, while the slopes of the x-axis and y-axis do not conform to this rule. Participants explore the definitions and calculations of slope, particularly in relation to undefined values and vertical lines.
Participants do not reach a consensus on the terminology used to describe the relationship between slopes of perpendicular lines. While some agree on the mathematical principles, others contest the phrasing of "slope between two lines" as misleading.
Limitations include the dependence on definitions of slope and the distinction between vertical and non-vertical lines. The discussion highlights the nuances in mathematical language and the implications of undefined values in slope calculations.
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?
please explain what is meant by hole in the graph?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).
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.BvU said:Slope between ? How do you calculate that ?
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 .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!
sort of sounded strange to me.Anukriti C. said:slope between two perpendicular lines is -1
And to me as well. You can talk about the angle between two perpendicular lines (which is 90°), but not the "slope" between themBvU said:The expressionsort of sounded strange to me.Anukriti C. said:slope between two perpendicular lines is -1
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.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
I believe that was a bit harsh way of correction.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.
Your question has pretty much been answered, but I'll sum things up.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.
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}##.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?
that was very much helpful.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.