The gradient of a line is a measure of how steep
that line is.
We may have a very steep line like
that.
And the gradient will be larger than a line.
Which is a bit more shallow.
So lines that are fairly shallow like this one will have fairly
Lopes, fairly small gradients,
steep lines. Have large gradients and lines.
Which are horizontal, will have zero gradients?
And we need to try to quantify that a little bit. Try and do
this a little bit more mathematically so we can
actually measure how much steeper this is than this one
than this one, and so on.
I'd run three line segments on this
diagram. Let's just look at them.
The first one.
Is the line segment from A to D?
Now as we move from A to D.
The X coordinate increases from one to two.
And the Y coordinate increases from one to five.
Now the line segment AD is steeper than the line segment
AC. As we move from A to C.
Exchange is from one to two and Y changes from one to three.
An AC in turn is steeper than the line segment AB. The line
segment AB is in fact horizontal because as X
increases from one to two, the Y coordinate doesn't change at
all. It remains at one.
Let's try and think about why mathematically, the line AD is
steeper than the line AC.
And the reason for this is that in both cases are X coordinate
is changing from one to two.
But as we move from A to D, there's a much bigger.
Change in Y than if we move from A to see so it's this relative
change in Y relative change in X, it's going to be important.
What we do is we calculate the change in Y.
Divide it by the change in X.
That's going to be a measure of the steepness. Let's do it for
the point AD for the points A&E.
OK, as we move from A to D.
Why changes from one to five? So the change in Y?
Is 5 - 1 and the change in X while exchanges from one to two.
So the change in X is 2 - 1.
So this quantity, the change in Y over change in X for the
line AD is 5 - 1, which is four 2 - 1 which is one and
four over one is 4.
So that's a measure of how much why changes as exchanges.
From one to two. What about the segment AC? Let's do
the same thing.
While the change in Y now is from one to three.
So the changes 3 - 1.
The change in X well X goes from one to two, so the changes 2 - 1
and again 3 - 1 is two. 2 - 1 is one and 2 / 1 is 2. So this
is a measure of the relative change in X&Y.
What about a bee?
Well, as we move from A to B, why doesn't change at all? So
the change in Y is 1 - 1, which is of course 0.
And the change in X is still 2 - 1, so we get zero over
one which is 0.
So you see this quantity change in Y divided by changing X gives
us a measure of the steepness of
these lines. As we would expect, the change in Y over change in X
for AD. Which we turned out to be 4 is greater than the change
in Y over change in X for AC because ady is steeper than AC
an intern. This change in Y over change in X for AC.
Is greater than the change in Y over change in X for a bee
because AC is steeper than a B?
So it's this quantity which gives us the measure that we're
looking for, and it's this quantity we define to be the
gradient of the line segment.
We often use the symbol M for
gradient. So the gradient is defined to be the change
in Y. Divided by the change in X.
As we move from one point to a
neighboring point. Let's do that for some general case.
Suppose we have.
System of coordinates
appoint a. X1Y One.
And a point B.
X2Y2 And we're interested in the gradient
of the line segment joining A&B.
Let me put in a horizontal line
through way. Anna vertical line.
Through be. So there's my X axis. There's my Y Axis.
As we move from A to B.
Exchange is from X one.
2X2.
Why changes from Y1?
To Y2.
So the change in Y divided by the change in X.
While the change in Y is the final value minus the initial
value, so it's Y 2 minus Y 1.
The change in X is X2 minus X one.
And that is the formula that we can always use to find the
gradient of the line joining two
points. We can think of this
another way. Suppose we look at this angle in here. Let's call
that angle theater.
Now the change in Y is just this distance here.
Distance in there.
And the change in X is this distance in here.
And if we take the change in Y and divide it by the change in
X, what we actually get is the ratio of this side of this right
angle triangle to this side. And that's just the tangent of this
angle here. So this quantity that we've calculated is not
only the gradient of the line, it's also the tangent
of the angle that the line makes with the horizontal.
So the gradient M which we said is Y 2 minus Y, one over X2
minus X one is also equal to the Tangent Theta, where Theta is
the angle that the line makes with the horizontal.
We can take this a stage further. Suppose we continue
this line backwards until we
meet. The X axis.
And this angle in here between the extended line and the X
axis. Corresponds to this angle. Here these are
corresponding angles, so this two must also be theater.
So In other words, the gradient of the line is also the tangent
of the angle that the line makes with the X axis.
Let's have a couple of examples.
Let's choose a couple of points. Supposing a is the .34.
And B is the point.
814. Let's calculate the gradient of the
line joining these two points.
Well, the gradient is simply the difference in the Y coordinates
14 - 4.
Over the difference in the X
coordinates. 8 - 3.
14 - 4 is 10 and 8. Subtract 3 is 5 and 5 into 10 goes
twice. So the gradient of this line is 2.
A second other example, suppose we have the point a, which
now has coordinates 04 and B which has coordinates 50. Let's
do the same calculation.
The gradient will be the difference in the Y coordinates.
That's 0 - 4.
Divided by the difference in the X coordinates 5 - 0.
So this time will get minus four on the top, five. At the bottom
we get minus four fifths.
So this is a little bit different now because we found
that we've got a negative number
for our gradient. And see what that actually means.
Let's plot the points and see what's going on.
Point A0X coordinate Y coordinate of four. So let's put
that there that's Point A.
Point B has an X coordinate of five that's there.
Y coordinate of 0, so there's my point there in there.
And the line joining them looks
like this. We know that this line has gradient
minus four fifths.
This line, as you notice, is sloping downwards as we move
from left to right.
And that's why the gradient turns out to be negative.
Another way of thinking about this is that the angle that the
line now makes with the X axis. This angle in here this theater
is now an obtuse angle.
Greater than 90 degrees less than 180 degrees, so we've an
obtuse angle. A line which is sloping downwards from left to
right. And a negative gradient.
Let me try to summarize all that behavior. If
you have a situation.
Like this?
Where the angle that the line makes with the
horizontal is acute.
Then the gradient.
Will be positive.
And the reason for that is that as you move along the line.
As X increases, why also increases so the change
in Y and the change in X have the same sign.
It's also important to recognize that if we take the tangent of
an acute angle, you get a positive number, so Tan Theater,
which we know is the same as them, is also positive.
What about an angle that sloping alignment sloping
downwards? We know that the angle is now.
Theater and it's obtuse.
We know that the tangent of an obtuse angle is negative, and as
we've seen, the gradient is
negative. And that's be'cause as X increases.
Why is decreasing so the change in Y and the change in X have
different signs, so we take the ratio, will find out that the
gradient is actually negative.
And finally.
Let's have one where theater is 0, so the angle that the line
makes with the horizontal is 0, while tan feta is 0. So that's
consistent with our intuition. That tells us that MSO the
gradients 0. Let's have a look
at some parallel
lines. Here's
a
line. Let's call it L1.
L1 will make a certain angle.
Theater one with the X axis.
So it's gradient, as we've seen already, is Tampa Theatre 1.
M1, its gradient is tan.
Theater.
Let's put another line on this, also parallel to this first
line. This line is L2.
It will have a gradient M2.
That's extend it back to the.
Horizontal axis And let's measure this angle that would be
theater 2. And M2 will be the tangent of Theta 2.
Now, because these two lines are parallel.
They cross this X axis at the same angle, Theta one and three
to two. A corresponding angles Sophie to one must be equal to
three to two.
So because the to one is 3 to 2.
10 three to one what he called 10 theater 2 so. In other words,
M1 equals M2.
So for two parallel lines, as you might have expected,
intuitively, the two gradients are equal.
And Conversely, if we have two lines for which the gradients
are equal. Then we can deduce from that that the two lines
must be parallel. OK, so that's parallel lines. Let's look at
some perpendicular lines.
See if we can do something about the gradients of
perpendicular lines.
Start with a point P.
And the origin there. And let's suppose point P has coordinates
a speed. That means that to get to pee from oh, we go a
distance a in the X direction and be in the Y direction.
Now what I'm going to do now is I'm going to draw a
perpendicular line, a line that is perpendicular to Opie, and
I'm going to do that by taking opian, rotating it through 90
degrees. So the point P will move around here and it will
move to appoint up there
somewhere. And let's call that
new point Q. I'm going to try to figure out what the
coordinates of QR.
This angle hit in here is 90 degrees because Opie and oq are
perpendicular. Now to get from O to pee, wee had to go
horizontally a distance A and vertically be. So if this
triangle shifts around over here to get from Otak you will have
to go vertically a distance a.
And then horizontally a distance
be. So you see, we've just shifted this triangle, rotated
it round through 90 degrees, and doing that we can then read off
the coordinates of Point Q.
Q will have an X coordinate of
minus B. And AY coordinate of A.
That's now calculate the gradient of the line opi. Let's
call that MOPMLP. Remember, is the change in Y divided by
changing X as we move from OTP.
As we move from outer P, the change in Y is B minus zero.
The change in X is a minus zero.
So the gradient of Opie is just be over A.
What about the gradient of OQ? Let's call that MOQ.
We want the change in Y divided by the change in X as we move
from O to Q.
Well, the change in Y as we move from outer Q is a subtract 0.
And the change in X is minus B, subtract 0.
So this time this simplifies to a over minus
B or minus a over B.
Now let's see what happens when we multiply these two results
together. Let's take MOP and we're going to multiply it by
MCU. That's be over a multiplied by minus a over B.
And you see, when we do that, the aids cancel the beast
cancel, and we're left with
just. Minus one.
This is a very important result if you have two perpendicular
lines, then the product of their gradients is always minus one.
And correspondingly, if you've got 2 lines and you find that
when you multiply the gradients together, you get minus one, you
can deduce from that that the lines must be perpendicular.
Let's just have a look at an
example. Let's have three
points. Using A is the
.12. Because the
.34. And see is the point is
not 3. And we'll ask ourselves, the question is AB.
Perpendicular
To AC. Question is a be perpendicular to AC?
Well will will do this by calculating the gradient of the
line from A to B. Let's call
that MAB. And then we'll find the gradients of the line from A
to C will call that Mac.
So let's do this calculation. We want the gradient of the line
from A to B. Well, that's simply the difference in the Y
coordinates 4 - 2.
Over the difference in the X coordinates 3 - 1.
4 - 2 is two 3 - 1 is 2, so the gradient of a B is one.
What about the gradient from A
to C? Well, the difference in the Y coordinates now is 3 - 2.
The difference in the X coordinates is 0 - 1.
So we've got 3 - 2 is one 0 - 1 is minus one.
So all this simplifies which is minus one.
If we multiply the two gradients together, maybe multiplied by
Mac, will get one times minus
one. Which is clearly minus one, so the gradients of these two
lines multiplied together have a result which is minus one, and
that means that the two lines a be an AC.
Indeed, must be perpendicular.