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.