[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.50,0:00:05.79,Default,,0000,0000,0000,,The gradient of a line is\Na measure of how steep
Dialogue: 0,0:00:05.79,0:00:06.96,Default,,0000,0000,0000,,that line is.
Dialogue: 0,0:00:11.06,0:00:18.27,Default,,0000,0000,0000,,We may have a\Nvery steep line like
Dialogue: 0,0:00:18.27,0:00:19.17,Default,,0000,0000,0000,,that.
Dialogue: 0,0:00:20.25,0:00:23.17,Default,,0000,0000,0000,,And the gradient will\Nbe larger than a line.
Dialogue: 0,0:00:24.59,0:00:26.33,Default,,0000,0000,0000,,Which is a bit more shallow.
Dialogue: 0,0:00:28.90,0:00:34.62,Default,,0000,0000,0000,,So lines that are fairly shallow\Nlike this one will have fairly
Dialogue: 0,0:00:34.62,0:00:36.53,Default,,0000,0000,0000,,Lopes, fairly small gradients,
Dialogue: 0,0:00:36.53,0:00:40.60,Default,,0000,0000,0000,,steep lines. Have large\Ngradients and lines.
Dialogue: 0,0:00:41.58,0:00:44.76,Default,,0000,0000,0000,,Which are horizontal, will\Nhave zero gradients?
Dialogue: 0,0:00:46.03,0:00:49.78,Default,,0000,0000,0000,,And we need to try to quantify\Nthat a little bit. Try and do
Dialogue: 0,0:00:49.78,0:00:52.19,Default,,0000,0000,0000,,this a little bit more\Nmathematically so we can
Dialogue: 0,0:00:52.19,0:00:54.87,Default,,0000,0000,0000,,actually measure how much\Nsteeper this is than this one
Dialogue: 0,0:00:54.87,0:00:56.48,Default,,0000,0000,0000,,than this one, and so on.
Dialogue: 0,0:00:57.13,0:01:02.77,Default,,0000,0000,0000,,I'd run three line\Nsegments on this
Dialogue: 0,0:01:02.77,0:01:07.61,Default,,0000,0000,0000,,diagram. Let's\Njust look at them.
Dialogue: 0,0:01:08.95,0:01:10.24,Default,,0000,0000,0000,,The first one.
Dialogue: 0,0:01:10.76,0:01:13.57,Default,,0000,0000,0000,,Is the line segment from A to D?
Dialogue: 0,0:01:14.92,0:01:17.32,Default,,0000,0000,0000,,Now as we move from A to D.
Dialogue: 0,0:01:18.08,0:01:21.45,Default,,0000,0000,0000,,The X coordinate increases from\None to two.
Dialogue: 0,0:01:23.70,0:01:26.72,Default,,0000,0000,0000,,And the Y coordinate increases\Nfrom one to five.
Dialogue: 0,0:01:27.53,0:01:33.52,Default,,0000,0000,0000,,Now the line segment AD is\Nsteeper than the line segment
Dialogue: 0,0:01:33.52,0:01:37.50,Default,,0000,0000,0000,,AC. As we move from A to C.
Dialogue: 0,0:01:38.69,0:01:43.23,Default,,0000,0000,0000,,Exchange is from one to two and\NY changes from one to three.
Dialogue: 0,0:01:44.56,0:01:49.19,Default,,0000,0000,0000,,An AC in turn is steeper than\Nthe line segment AB. The line
Dialogue: 0,0:01:49.19,0:01:52.39,Default,,0000,0000,0000,,segment AB is in fact\Nhorizontal because as X
Dialogue: 0,0:01:52.39,0:01:56.31,Default,,0000,0000,0000,,increases from one to two, the\NY coordinate doesn't change at
Dialogue: 0,0:01:56.31,0:01:58.09,Default,,0000,0000,0000,,all. It remains at one.
Dialogue: 0,0:01:59.33,0:02:04.42,Default,,0000,0000,0000,,Let's try and think about why\Nmathematically, the line AD is
Dialogue: 0,0:02:04.42,0:02:06.74,Default,,0000,0000,0000,,steeper than the line AC.
Dialogue: 0,0:02:07.65,0:02:12.17,Default,,0000,0000,0000,,And the reason for this is that\Nin both cases are X coordinate
Dialogue: 0,0:02:12.17,0:02:14.26,Default,,0000,0000,0000,,is changing from one to two.
Dialogue: 0,0:02:15.21,0:02:18.63,Default,,0000,0000,0000,,But as we move from A to D,\Nthere's a much bigger.
Dialogue: 0,0:02:19.13,0:02:24.35,Default,,0000,0000,0000,,Change in Y than if we move from\NA to see so it's this relative
Dialogue: 0,0:02:24.35,0:02:28.53,Default,,0000,0000,0000,,change in Y relative change in\NX, it's going to be important.
Dialogue: 0,0:02:29.08,0:02:34.22,Default,,0000,0000,0000,,What we do is we calculate the\Nchange in Y.
Dialogue: 0,0:02:34.24,0:02:40.29,Default,,0000,0000,0000,,Divide it by the\Nchange in X.
Dialogue: 0,0:02:40.30,0:02:46.60,Default,,0000,0000,0000,,That's going to be a measure of\Nthe steepness. Let's do it for
Dialogue: 0,0:02:46.60,0:02:50.00,Default,,0000,0000,0000,,the point AD for the points A&E.
Dialogue: 0,0:02:50.07,0:02:52.96,Default,,0000,0000,0000,,OK, as we move from A to D.
Dialogue: 0,0:02:54.07,0:02:58.83,Default,,0000,0000,0000,,Why changes from one to five? So\Nthe change in Y?
Dialogue: 0,0:02:59.43,0:03:04.18,Default,,0000,0000,0000,,Is 5 - 1 and the change in X\Nwhile exchanges from one to two.
Dialogue: 0,0:03:04.18,0:03:07.04,Default,,0000,0000,0000,,So the change in X is 2 - 1.
Dialogue: 0,0:03:07.60,0:03:11.85,Default,,0000,0000,0000,,So this quantity, the change\Nin Y over change in X for the
Dialogue: 0,0:03:11.85,0:03:17.08,Default,,0000,0000,0000,,line AD is 5 - 1, which is\Nfour 2 - 1 which is one and
Dialogue: 0,0:03:17.08,0:03:18.72,Default,,0000,0000,0000,,four over one is 4.
Dialogue: 0,0:03:20.50,0:03:24.80,Default,,0000,0000,0000,,So that's a measure of how much\Nwhy changes as exchanges.
Dialogue: 0,0:03:26.02,0:03:30.42,Default,,0000,0000,0000,,From one to two. What about\Nthe segment AC? Let's do
Dialogue: 0,0:03:30.42,0:03:31.59,Default,,0000,0000,0000,,the same thing.
Dialogue: 0,0:03:32.88,0:03:36.86,Default,,0000,0000,0000,,While the change in Y now is\Nfrom one to three.
Dialogue: 0,0:03:37.82,0:03:40.59,Default,,0000,0000,0000,,So the changes 3 - 1.
Dialogue: 0,0:03:42.93,0:03:49.39,Default,,0000,0000,0000,,The change in X well X goes from\None to two, so the changes 2 - 1
Dialogue: 0,0:03:49.39,0:03:56.99,Default,,0000,0000,0000,,and again 3 - 1 is two. 2 - 1\Nis one and 2 / 1 is 2. So this
Dialogue: 0,0:03:56.99,0:04:00.41,Default,,0000,0000,0000,,is a measure of the relative\Nchange in X&Y.
Dialogue: 0,0:04:01.64,0:04:04.45,Default,,0000,0000,0000,,What about a bee?
Dialogue: 0,0:04:04.45,0:04:09.28,Default,,0000,0000,0000,,Well, as we move from A to B,\Nwhy doesn't change at all? So
Dialogue: 0,0:04:09.28,0:04:13.76,Default,,0000,0000,0000,,the change in Y is 1 - 1, which\Nis of course 0.
Dialogue: 0,0:04:14.29,0:04:19.36,Default,,0000,0000,0000,,And the change in X is still\N2 - 1, so we get zero over
Dialogue: 0,0:04:19.36,0:04:20.71,Default,,0000,0000,0000,,one which is 0.
Dialogue: 0,0:04:22.24,0:04:26.66,Default,,0000,0000,0000,,So you see this quantity change\Nin Y divided by changing X gives
Dialogue: 0,0:04:26.66,0:04:29.04,Default,,0000,0000,0000,,us a measure of the steepness of
Dialogue: 0,0:04:29.04,0:04:33.86,Default,,0000,0000,0000,,these lines. As we would expect,\Nthe change in Y over change in X
Dialogue: 0,0:04:33.86,0:04:39.08,Default,,0000,0000,0000,,for AD. Which we turned out to\Nbe 4 is greater than the change
Dialogue: 0,0:04:39.08,0:04:43.86,Default,,0000,0000,0000,,in Y over change in X for AC\Nbecause ady is steeper than AC
Dialogue: 0,0:04:43.86,0:04:48.24,Default,,0000,0000,0000,,an intern. This change in Y over\Nchange in X for AC.
Dialogue: 0,0:04:48.74,0:04:53.01,Default,,0000,0000,0000,,Is greater than the change in Y\Nover change in X for a bee
Dialogue: 0,0:04:53.01,0:04:55.14,Default,,0000,0000,0000,,because AC is steeper than a B?
Dialogue: 0,0:04:55.73,0:04:59.60,Default,,0000,0000,0000,,So it's this quantity which\Ngives us the measure that we're
Dialogue: 0,0:04:59.60,0:05:03.47,Default,,0000,0000,0000,,looking for, and it's this\Nquantity we define to be the
Dialogue: 0,0:05:03.47,0:05:05.23,Default,,0000,0000,0000,,gradient of the line segment.
Dialogue: 0,0:05:06.65,0:05:10.58,Default,,0000,0000,0000,,We often use the symbol M for
Dialogue: 0,0:05:10.58,0:05:17.90,Default,,0000,0000,0000,,gradient. So the gradient is\Ndefined to be the change
Dialogue: 0,0:05:17.90,0:05:24.83,Default,,0000,0000,0000,,in Y. Divided by\Nthe change in X.
Dialogue: 0,0:05:24.83,0:05:27.78,Default,,0000,0000,0000,,As we move from one point to a
Dialogue: 0,0:05:27.78,0:05:35.12,Default,,0000,0000,0000,,neighboring point. Let's do that\Nfor some general case.
Dialogue: 0,0:05:35.12,0:05:37.72,Default,,0000,0000,0000,,Suppose we have.
Dialogue: 0,0:05:38.37,0:05:42.05,Default,,0000,0000,0000,,System of coordinates
Dialogue: 0,0:05:42.05,0:05:46.22,Default,,0000,0000,0000,,appoint a. X1Y\NOne.
Dialogue: 0,0:05:48.29,0:05:50.66,Default,,0000,0000,0000,,And a point B.
Dialogue: 0,0:05:50.66,0:05:56.78,Default,,0000,0000,0000,,X2Y2 And\Nwe're interested in the gradient
Dialogue: 0,0:05:56.78,0:05:59.42,Default,,0000,0000,0000,,of the line segment joining A&B.
Dialogue: 0,0:06:00.51,0:06:04.12,Default,,0000,0000,0000,,Let me put in a horizontal line
Dialogue: 0,0:06:04.12,0:06:06.91,Default,,0000,0000,0000,,through way. Anna vertical line.
Dialogue: 0,0:06:07.58,0:06:13.21,Default,,0000,0000,0000,,Through be. So there's my X\Naxis. There's my Y Axis.
Dialogue: 0,0:06:13.97,0:06:17.22,Default,,0000,0000,0000,,As we move from A to B.
Dialogue: 0,0:06:17.76,0:06:20.36,Default,,0000,0000,0000,,Exchange is from X one.
Dialogue: 0,0:06:20.93,0:06:22.26,Default,,0000,0000,0000,,2X2.
Dialogue: 0,0:06:24.23,0:06:26.30,Default,,0000,0000,0000,,Why changes from Y1?
Dialogue: 0,0:06:26.80,0:06:27.64,Default,,0000,0000,0000,,To Y2.
Dialogue: 0,0:06:29.39,0:06:35.90,Default,,0000,0000,0000,,So the change in Y divided\Nby the change in X.
Dialogue: 0,0:06:36.46,0:06:42.44,Default,,0000,0000,0000,,While the change in Y is the\Nfinal value minus the initial
Dialogue: 0,0:06:42.44,0:06:46.42,Default,,0000,0000,0000,,value, so it's Y 2 minus Y 1.
Dialogue: 0,0:06:46.63,0:06:51.81,Default,,0000,0000,0000,,The change in X is X2\Nminus X one.
Dialogue: 0,0:06:51.82,0:06:57.45,Default,,0000,0000,0000,,And that is the formula that we\Ncan always use to find the
Dialogue: 0,0:06:57.45,0:07:00.05,Default,,0000,0000,0000,,gradient of the line joining two
Dialogue: 0,0:07:00.05,0:07:03.34,Default,,0000,0000,0000,,points. We can think of this
Dialogue: 0,0:07:03.34,0:07:08.96,Default,,0000,0000,0000,,another way. Suppose we look at\Nthis angle in here. Let's call
Dialogue: 0,0:07:08.96,0:07:10.11,Default,,0000,0000,0000,,that angle theater.
Dialogue: 0,0:07:11.50,0:07:14.71,Default,,0000,0000,0000,,Now the change in Y is\Njust this distance here.
Dialogue: 0,0:07:16.23,0:07:18.84,Default,,0000,0000,0000,,Distance in there.
Dialogue: 0,0:07:18.84,0:07:23.04,Default,,0000,0000,0000,,And the change in X is this\Ndistance in here.
Dialogue: 0,0:07:24.13,0:07:28.22,Default,,0000,0000,0000,,And if we take the change in Y\Nand divide it by the change in
Dialogue: 0,0:07:28.22,0:07:32.05,Default,,0000,0000,0000,,X, what we actually get is the\Nratio of this side of this right
Dialogue: 0,0:07:32.05,0:07:35.32,Default,,0000,0000,0000,,angle triangle to this side. And\Nthat's just the tangent of this
Dialogue: 0,0:07:35.32,0:07:39.48,Default,,0000,0000,0000,,angle here. So this quantity\Nthat we've calculated is not
Dialogue: 0,0:07:39.48,0:07:42.99,Default,,0000,0000,0000,,only the gradient of the\Nline, it's also the tangent
Dialogue: 0,0:07:42.99,0:07:46.50,Default,,0000,0000,0000,,of the angle that the line\Nmakes with the horizontal.
Dialogue: 0,0:07:47.76,0:07:54.18,Default,,0000,0000,0000,,So the gradient M which we said\Nis Y 2 minus Y, one over X2
Dialogue: 0,0:07:54.18,0:07:59.74,Default,,0000,0000,0000,,minus X one is also equal to the\NTangent Theta, where Theta is
Dialogue: 0,0:07:59.74,0:08:03.60,Default,,0000,0000,0000,,the angle that the line makes\Nwith the horizontal.
Dialogue: 0,0:08:05.57,0:08:09.60,Default,,0000,0000,0000,,We can take this a stage\Nfurther. Suppose we continue
Dialogue: 0,0:08:09.60,0:08:11.62,Default,,0000,0000,0000,,this line backwards until we
Dialogue: 0,0:08:11.62,0:08:13.66,Default,,0000,0000,0000,,meet. The X axis.
Dialogue: 0,0:08:14.47,0:08:19.20,Default,,0000,0000,0000,,And this angle in here between\Nthe extended line and the X
Dialogue: 0,0:08:19.20,0:08:22.47,Default,,0000,0000,0000,,axis. Corresponds to this\Nangle. Here these are
Dialogue: 0,0:08:22.47,0:08:25.15,Default,,0000,0000,0000,,corresponding angles, so this\Ntwo must also be theater.
Dialogue: 0,0:08:26.89,0:08:30.97,Default,,0000,0000,0000,,So In other words, the gradient\Nof the line is also the tangent
Dialogue: 0,0:08:30.97,0:08:34.43,Default,,0000,0000,0000,,of the angle that the line makes\Nwith the X axis.
Dialogue: 0,0:08:35.88,0:08:42.35,Default,,0000,0000,0000,,Let's have a\Ncouple of examples.
Dialogue: 0,0:08:42.94,0:08:48.78,Default,,0000,0000,0000,,Let's choose a couple of points.\NSupposing a is the .34.
Dialogue: 0,0:08:48.79,0:08:50.74,Default,,0000,0000,0000,,And B is the point.
Dialogue: 0,0:08:51.55,0:08:58.05,Default,,0000,0000,0000,,814. Let's\Ncalculate the gradient of the
Dialogue: 0,0:08:58.05,0:09:01.15,Default,,0000,0000,0000,,line joining these two points.
Dialogue: 0,0:09:01.16,0:09:06.88,Default,,0000,0000,0000,,Well, the gradient is simply the\Ndifference in the Y coordinates
Dialogue: 0,0:09:06.88,0:09:08.44,Default,,0000,0000,0000,,14 - 4.
Dialogue: 0,0:09:08.50,0:09:10.74,Default,,0000,0000,0000,,Over the difference in the X
Dialogue: 0,0:09:10.74,0:09:14.37,Default,,0000,0000,0000,,coordinates. 8 - 3.
Dialogue: 0,0:09:14.37,0:09:21.83,Default,,0000,0000,0000,,14 - 4 is 10 and 8. Subtract\N3 is 5 and 5 into 10 goes
Dialogue: 0,0:09:21.83,0:09:26.02,Default,,0000,0000,0000,,twice. So the gradient of this\Nline is 2.
Dialogue: 0,0:09:26.70,0:09:33.51,Default,,0000,0000,0000,,A second other example, suppose\Nwe have the point a, which
Dialogue: 0,0:09:33.51,0:09:40.32,Default,,0000,0000,0000,,now has coordinates 04 and B\Nwhich has coordinates 50. Let's
Dialogue: 0,0:09:40.32,0:09:42.79,Default,,0000,0000,0000,,do the same calculation.
Dialogue: 0,0:09:42.81,0:09:47.14,Default,,0000,0000,0000,,The gradient will be the\Ndifference in the Y coordinates.
Dialogue: 0,0:09:47.14,0:09:48.87,Default,,0000,0000,0000,,That's 0 - 4.
Dialogue: 0,0:09:48.88,0:09:54.51,Default,,0000,0000,0000,,Divided by the difference in the\NX coordinates 5 - 0.
Dialogue: 0,0:09:54.57,0:09:59.29,Default,,0000,0000,0000,,So this time will get minus four\Non the top, five. At the bottom
Dialogue: 0,0:09:59.29,0:10:00.97,Default,,0000,0000,0000,,we get minus four fifths.
Dialogue: 0,0:10:02.04,0:10:05.22,Default,,0000,0000,0000,,So this is a little bit\Ndifferent now because we found
Dialogue: 0,0:10:05.22,0:10:06.95,Default,,0000,0000,0000,,that we've got a negative number
Dialogue: 0,0:10:06.95,0:10:10.31,Default,,0000,0000,0000,,for our gradient. And see\Nwhat that actually means.
Dialogue: 0,0:10:10.31,0:10:12.80,Default,,0000,0000,0000,,Let's plot the points and\Nsee what's going on.
Dialogue: 0,0:10:14.96,0:10:21.27,Default,,0000,0000,0000,,Point A0X coordinate Y\Ncoordinate of four. So let's put
Dialogue: 0,0:10:21.27,0:10:24.42,Default,,0000,0000,0000,,that there that's Point A.
Dialogue: 0,0:10:25.23,0:10:28.84,Default,,0000,0000,0000,,Point B has an X coordinate of\Nfive that's there.
Dialogue: 0,0:10:29.37,0:10:33.99,Default,,0000,0000,0000,,Y coordinate of 0, so there's my\Npoint there in there.
Dialogue: 0,0:10:34.53,0:10:37.51,Default,,0000,0000,0000,,And the line joining them looks
Dialogue: 0,0:10:37.51,0:10:42.20,Default,,0000,0000,0000,,like this. We know that\Nthis line has gradient
Dialogue: 0,0:10:42.20,0:10:43.54,Default,,0000,0000,0000,,minus four fifths.
Dialogue: 0,0:10:44.81,0:10:48.44,Default,,0000,0000,0000,,This line, as you notice, is\Nsloping downwards as we move
Dialogue: 0,0:10:48.44,0:10:49.76,Default,,0000,0000,0000,,from left to right.
Dialogue: 0,0:10:50.63,0:10:54.27,Default,,0000,0000,0000,,And that's why the gradient\Nturns out to be negative.
Dialogue: 0,0:10:55.35,0:10:59.65,Default,,0000,0000,0000,,Another way of thinking about\Nthis is that the angle that the
Dialogue: 0,0:10:59.65,0:11:04.30,Default,,0000,0000,0000,,line now makes with the X axis.\NThis angle in here this theater
Dialogue: 0,0:11:04.30,0:11:06.09,Default,,0000,0000,0000,,is now an obtuse angle.
Dialogue: 0,0:11:06.17,0:11:10.46,Default,,0000,0000,0000,,Greater than 90 degrees less\Nthan 180 degrees, so we've an
Dialogue: 0,0:11:10.46,0:11:14.70,Default,,0000,0000,0000,,obtuse angle. A line which is\Nsloping downwards from left to
Dialogue: 0,0:11:14.70,0:11:17.63,Default,,0000,0000,0000,,right. And a negative gradient.
Dialogue: 0,0:11:18.98,0:11:24.49,Default,,0000,0000,0000,,Let me try to summarize\Nall that behavior. If
Dialogue: 0,0:11:24.49,0:11:26.94,Default,,0000,0000,0000,,you have a situation.
Dialogue: 0,0:11:28.15,0:11:30.90,Default,,0000,0000,0000,,Like this?
Dialogue: 0,0:11:32.08,0:11:37.08,Default,,0000,0000,0000,,Where the angle that the\Nline makes with the
Dialogue: 0,0:11:37.08,0:11:38.74,Default,,0000,0000,0000,,horizontal is acute.
Dialogue: 0,0:11:40.33,0:11:42.37,Default,,0000,0000,0000,,Then the gradient.
Dialogue: 0,0:11:42.99,0:11:44.82,Default,,0000,0000,0000,,Will be positive.
Dialogue: 0,0:11:45.56,0:11:49.29,Default,,0000,0000,0000,,And the reason for that is that\Nas you move along the line.
Dialogue: 0,0:11:49.90,0:11:53.02,Default,,0000,0000,0000,,As X increases, why also\Nincreases so the change
Dialogue: 0,0:11:53.02,0:11:56.84,Default,,0000,0000,0000,,in Y and the change in X\Nhave the same sign.
Dialogue: 0,0:11:58.64,0:12:02.00,Default,,0000,0000,0000,,It's also important to recognize\Nthat if we take the tangent of
Dialogue: 0,0:12:02.00,0:12:05.08,Default,,0000,0000,0000,,an acute angle, you get a\Npositive number, so Tan Theater,
Dialogue: 0,0:12:05.08,0:12:08.16,Default,,0000,0000,0000,,which we know is the same as\Nthem, is also positive.
Dialogue: 0,0:12:10.12,0:12:16.34,Default,,0000,0000,0000,,What about an angle that\Nsloping alignment sloping
Dialogue: 0,0:12:16.34,0:12:21.02,Default,,0000,0000,0000,,downwards? We know that the\Nangle is now.
Dialogue: 0,0:12:21.59,0:12:23.56,Default,,0000,0000,0000,,Theater and it's obtuse.
Dialogue: 0,0:12:24.70,0:12:29.86,Default,,0000,0000,0000,,We know that the tangent of an\Nobtuse angle is negative, and as
Dialogue: 0,0:12:29.86,0:12:31.85,Default,,0000,0000,0000,,we've seen, the gradient is
Dialogue: 0,0:12:31.85,0:12:37.55,Default,,0000,0000,0000,,negative. And that's be'cause\Nas X increases.
Dialogue: 0,0:12:38.30,0:12:43.02,Default,,0000,0000,0000,,Why is decreasing so the change\Nin Y and the change in X have
Dialogue: 0,0:12:43.02,0:12:47.06,Default,,0000,0000,0000,,different signs, so we take the\Nratio, will find out that the
Dialogue: 0,0:12:47.06,0:12:48.41,Default,,0000,0000,0000,,gradient is actually negative.
Dialogue: 0,0:12:49.68,0:12:50.73,Default,,0000,0000,0000,,And finally.
Dialogue: 0,0:12:51.78,0:12:57.04,Default,,0000,0000,0000,,Let's have one where theater is\N0, so the angle that the line
Dialogue: 0,0:12:57.04,0:13:02.31,Default,,0000,0000,0000,,makes with the horizontal is 0,\Nwhile tan feta is 0. So that's
Dialogue: 0,0:13:02.31,0:13:06.36,Default,,0000,0000,0000,,consistent with our intuition.\NThat tells us that MSO the
Dialogue: 0,0:13:06.36,0:13:12.91,Default,,0000,0000,0000,,gradients 0. Let's\Nhave a look
Dialogue: 0,0:13:12.91,0:13:16.50,Default,,0000,0000,0000,,at some parallel
Dialogue: 0,0:13:16.50,0:13:23.42,Default,,0000,0000,0000,,lines.\NHere's
Dialogue: 0,0:13:23.42,0:13:29.13,Default,,0000,0000,0000,,a
Dialogue: 0,0:13:29.13,0:13:36.98,Default,,0000,0000,0000,,line.\NLet's call it L1.
Dialogue: 0,0:13:37.94,0:13:41.85,Default,,0000,0000,0000,,L1 will make a certain angle.
Dialogue: 0,0:13:42.08,0:13:44.29,Default,,0000,0000,0000,,Theater one with the X axis.
Dialogue: 0,0:13:44.98,0:13:50.38,Default,,0000,0000,0000,,So it's gradient, as we've seen\Nalready, is Tampa Theatre 1.
Dialogue: 0,0:13:51.00,0:13:54.49,Default,,0000,0000,0000,,M1, its gradient is tan.
Dialogue: 0,0:13:55.17,0:13:56.37,Default,,0000,0000,0000,,Theater.
Dialogue: 0,0:13:57.42,0:14:00.30,Default,,0000,0000,0000,,Let's put another line on this,\Nalso parallel to this first
Dialogue: 0,0:14:00.30,0:14:03.80,Default,,0000,0000,0000,,line. This line is L2.
Dialogue: 0,0:14:04.62,0:14:06.94,Default,,0000,0000,0000,,It will have a gradient M2.
Dialogue: 0,0:14:07.62,0:14:09.59,Default,,0000,0000,0000,,That's extend it back to the.
Dialogue: 0,0:14:10.40,0:14:14.41,Default,,0000,0000,0000,,Horizontal axis And let's\Nmeasure this angle that would be
Dialogue: 0,0:14:14.41,0:14:19.60,Default,,0000,0000,0000,,theater 2. And M2 will be the\Ntangent of Theta 2.
Dialogue: 0,0:14:20.95,0:14:25.11,Default,,0000,0000,0000,,Now, because these two\Nlines are parallel.
Dialogue: 0,0:14:26.51,0:14:30.90,Default,,0000,0000,0000,,They cross this X axis at the\Nsame angle, Theta one and three
Dialogue: 0,0:14:30.90,0:14:34.96,Default,,0000,0000,0000,,to two. A corresponding angles\NSophie to one must be equal to
Dialogue: 0,0:14:34.96,0:14:35.97,Default,,0000,0000,0000,,three to two.
Dialogue: 0,0:14:36.59,0:14:39.81,Default,,0000,0000,0000,,So because the to one is 3 to 2.
Dialogue: 0,0:14:40.44,0:14:45.76,Default,,0000,0000,0000,,10 three to one what he called\N10 theater 2 so. In other words,
Dialogue: 0,0:14:45.76,0:14:46.90,Default,,0000,0000,0000,,M1 equals M2.
Dialogue: 0,0:14:46.90,0:14:52.91,Default,,0000,0000,0000,,So for two parallel lines,\Nas you might have expected,
Dialogue: 0,0:14:52.91,0:14:56.52,Default,,0000,0000,0000,,intuitively, the two\Ngradients are equal.
Dialogue: 0,0:14:57.85,0:15:01.28,Default,,0000,0000,0000,,And Conversely, if we have two\Nlines for which the gradients
Dialogue: 0,0:15:01.28,0:15:05.57,Default,,0000,0000,0000,,are equal. Then we can deduce\Nfrom that that the two lines
Dialogue: 0,0:15:05.57,0:15:12.33,Default,,0000,0000,0000,,must be parallel. OK, so that's\Nparallel lines. Let's look at
Dialogue: 0,0:15:12.33,0:15:14.25,Default,,0000,0000,0000,,some perpendicular lines.
Dialogue: 0,0:15:15.26,0:15:18.21,Default,,0000,0000,0000,,See if we can do something\Nabout the gradients of
Dialogue: 0,0:15:18.21,0:15:18.80,Default,,0000,0000,0000,,perpendicular lines.
Dialogue: 0,0:15:20.65,0:15:23.20,Default,,0000,0000,0000,,Start with a point P.
Dialogue: 0,0:15:23.79,0:15:29.68,Default,,0000,0000,0000,,And the origin there. And let's\Nsuppose point P has coordinates
Dialogue: 0,0:15:29.68,0:15:37.16,Default,,0000,0000,0000,,a speed. That means that to get\Nto pee from oh, we go a
Dialogue: 0,0:15:37.16,0:15:43.58,Default,,0000,0000,0000,,distance a in the X direction\Nand be in the Y direction.
Dialogue: 0,0:15:44.51,0:15:48.10,Default,,0000,0000,0000,,Now what I'm going to do now is\NI'm going to draw a
Dialogue: 0,0:15:48.10,0:15:50.86,Default,,0000,0000,0000,,perpendicular line, a line that\Nis perpendicular to Opie, and
Dialogue: 0,0:15:50.86,0:15:54.17,Default,,0000,0000,0000,,I'm going to do that by taking\Nopian, rotating it through 90
Dialogue: 0,0:15:54.17,0:15:57.48,Default,,0000,0000,0000,,degrees. So the point P will\Nmove around here and it will
Dialogue: 0,0:15:57.48,0:15:58.86,Default,,0000,0000,0000,,move to appoint up there
Dialogue: 0,0:15:58.86,0:16:00.92,Default,,0000,0000,0000,,somewhere. And let's call that
Dialogue: 0,0:16:00.92,0:16:04.50,Default,,0000,0000,0000,,new point Q. I'm going to\Ntry to figure out what the
Dialogue: 0,0:16:04.50,0:16:05.24,Default,,0000,0000,0000,,coordinates of QR.
Dialogue: 0,0:16:06.78,0:16:12.03,Default,,0000,0000,0000,,This angle hit in here is 90\Ndegrees because Opie and oq are
Dialogue: 0,0:16:12.03,0:16:17.14,Default,,0000,0000,0000,,perpendicular. Now to get from O\Nto pee, wee had to go
Dialogue: 0,0:16:17.14,0:16:20.85,Default,,0000,0000,0000,,horizontally a distance A and\Nvertically be. So if this
Dialogue: 0,0:16:20.85,0:16:25.30,Default,,0000,0000,0000,,triangle shifts around over here\Nto get from Otak you will have
Dialogue: 0,0:16:25.30,0:16:27.53,Default,,0000,0000,0000,,to go vertically a distance a.
Dialogue: 0,0:16:30.80,0:16:32.50,Default,,0000,0000,0000,,And then horizontally a distance
Dialogue: 0,0:16:32.50,0:16:36.13,Default,,0000,0000,0000,,be. So you see, we've just\Nshifted this triangle, rotated
Dialogue: 0,0:16:36.13,0:16:40.12,Default,,0000,0000,0000,,it round through 90 degrees, and\Ndoing that we can then read off
Dialogue: 0,0:16:40.12,0:16:41.66,Default,,0000,0000,0000,,the coordinates of Point Q.
Dialogue: 0,0:16:42.36,0:16:44.67,Default,,0000,0000,0000,,Q will have an X coordinate of
Dialogue: 0,0:16:44.67,0:16:50.88,Default,,0000,0000,0000,,minus B. And AY\Ncoordinate of A.
Dialogue: 0,0:16:51.67,0:16:56.70,Default,,0000,0000,0000,,That's now calculate the\Ngradient of the line opi. Let's
Dialogue: 0,0:16:56.70,0:17:02.23,Default,,0000,0000,0000,,call that MOPMLP. Remember, is\Nthe change in Y divided by
Dialogue: 0,0:17:02.23,0:17:05.75,Default,,0000,0000,0000,,changing X as we move from OTP.
Dialogue: 0,0:17:06.37,0:17:12.67,Default,,0000,0000,0000,,As we move from outer P, the\Nchange in Y is B minus zero.
Dialogue: 0,0:17:12.67,0:17:18.17,Default,,0000,0000,0000,,The change in X is\Na minus zero.
Dialogue: 0,0:17:18.17,0:17:24.34,Default,,0000,0000,0000,,So the gradient of Opie is\Njust be over A.
Dialogue: 0,0:17:24.35,0:17:31.40,Default,,0000,0000,0000,,What about the gradient of\NOQ? Let's call that MOQ.
Dialogue: 0,0:17:33.04,0:17:36.91,Default,,0000,0000,0000,,We want the change in Y divided\Nby the change in X as we move
Dialogue: 0,0:17:36.91,0:17:37.94,Default,,0000,0000,0000,,from O to Q.
Dialogue: 0,0:17:39.06,0:17:46.23,Default,,0000,0000,0000,,Well, the change in Y as we move\Nfrom outer Q is a subtract 0.
Dialogue: 0,0:17:46.35,0:17:51.90,Default,,0000,0000,0000,,And the change in X is minus\NB, subtract 0.
Dialogue: 0,0:17:51.90,0:17:55.98,Default,,0000,0000,0000,,So this time this\Nsimplifies to a over minus
Dialogue: 0,0:17:55.98,0:17:58.70,Default,,0000,0000,0000,,B or minus a over B.
Dialogue: 0,0:18:00.78,0:18:05.30,Default,,0000,0000,0000,,Now let's see what happens when\Nwe multiply these two results
Dialogue: 0,0:18:05.30,0:18:09.82,Default,,0000,0000,0000,,together. Let's take MOP and\Nwe're going to multiply it by
Dialogue: 0,0:18:09.82,0:18:17.20,Default,,0000,0000,0000,,MCU. That's be over a\Nmultiplied by minus a over B.
Dialogue: 0,0:18:17.74,0:18:22.49,Default,,0000,0000,0000,,And you see, when we do that,\Nthe aids cancel the beast
Dialogue: 0,0:18:22.49,0:18:24.47,Default,,0000,0000,0000,,cancel, and we're left with
Dialogue: 0,0:18:24.47,0:18:25.70,Default,,0000,0000,0000,,just. Minus one.
Dialogue: 0,0:18:26.82,0:18:31.28,Default,,0000,0000,0000,,This is a very important result\Nif you have two perpendicular
Dialogue: 0,0:18:31.28,0:18:35.73,Default,,0000,0000,0000,,lines, then the product of their\Ngradients is always minus one.
Dialogue: 0,0:18:36.90,0:18:40.12,Default,,0000,0000,0000,,And correspondingly, if you've\Ngot 2 lines and you find that
Dialogue: 0,0:18:40.12,0:18:43.35,Default,,0000,0000,0000,,when you multiply the gradients\Ntogether, you get minus one, you
Dialogue: 0,0:18:43.35,0:18:46.28,Default,,0000,0000,0000,,can deduce from that that the\Nlines must be perpendicular.
Dialogue: 0,0:18:47.72,0:18:50.91,Default,,0000,0000,0000,,Let's just have a look at an
Dialogue: 0,0:18:50.91,0:18:53.70,Default,,0000,0000,0000,,example. Let's have three
Dialogue: 0,0:18:53.70,0:18:57.10,Default,,0000,0000,0000,,points. Using A is the
Dialogue: 0,0:18:57.10,0:19:00.53,Default,,0000,0000,0000,,.12. Because the
Dialogue: 0,0:19:00.53,0:19:04.53,Default,,0000,0000,0000,,.34. And see is the point is
Dialogue: 0,0:19:04.53,0:19:10.52,Default,,0000,0000,0000,,not 3. And we'll ask ourselves,\Nthe question is AB.
Dialogue: 0,0:19:10.52,0:19:11.46,Default,,0000,0000,0000,,Perpendicular
Dialogue: 0,0:19:12.53,0:19:17.82,Default,,0000,0000,0000,,To AC. Question is a\Nbe perpendicular to AC?
Dialogue: 0,0:19:18.48,0:19:23.36,Default,,0000,0000,0000,,Well will will do this by\Ncalculating the gradient of the
Dialogue: 0,0:19:23.36,0:19:26.47,Default,,0000,0000,0000,,line from A to B. Let's call
Dialogue: 0,0:19:26.47,0:19:31.07,Default,,0000,0000,0000,,that MAB. And then we'll find\Nthe gradients of the line from A
Dialogue: 0,0:19:31.07,0:19:32.92,Default,,0000,0000,0000,,to C will call that Mac.
Dialogue: 0,0:19:33.49,0:19:37.59,Default,,0000,0000,0000,,So let's do this calculation. We\Nwant the gradient of the line
Dialogue: 0,0:19:37.59,0:19:41.70,Default,,0000,0000,0000,,from A to B. Well, that's simply\Nthe difference in the Y
Dialogue: 0,0:19:41.70,0:19:43.07,Default,,0000,0000,0000,,coordinates 4 - 2.
Dialogue: 0,0:19:43.11,0:19:47.20,Default,,0000,0000,0000,,Over the difference in the X\Ncoordinates 3 - 1.
Dialogue: 0,0:19:47.20,0:19:53.52,Default,,0000,0000,0000,,4 - 2 is two 3 - 1 is 2, so\Nthe gradient of a B is one.
Dialogue: 0,0:19:54.88,0:19:56.52,Default,,0000,0000,0000,,What about the gradient from A
Dialogue: 0,0:19:56.52,0:20:02.97,Default,,0000,0000,0000,,to C? Well, the difference in\Nthe Y coordinates now is 3 - 2.
Dialogue: 0,0:20:03.00,0:20:07.48,Default,,0000,0000,0000,,The difference in the X\Ncoordinates is 0 - 1.
Dialogue: 0,0:20:07.81,0:20:14.53,Default,,0000,0000,0000,,So we've got 3 - 2 is one\N0 - 1 is minus one.
Dialogue: 0,0:20:14.54,0:20:16.92,Default,,0000,0000,0000,,So all this simplifies\Nwhich is minus one.
Dialogue: 0,0:20:18.23,0:20:23.28,Default,,0000,0000,0000,,If we multiply the two gradients\Ntogether, maybe multiplied by
Dialogue: 0,0:20:23.28,0:20:26.31,Default,,0000,0000,0000,,Mac, will get one times minus
Dialogue: 0,0:20:26.31,0:20:31.33,Default,,0000,0000,0000,,one. Which is clearly minus one,\Nso the gradients of these two
Dialogue: 0,0:20:31.33,0:20:35.82,Default,,0000,0000,0000,,lines multiplied together have a\Nresult which is minus one, and
Dialogue: 0,0:20:35.82,0:20:39.90,Default,,0000,0000,0000,,that means that the two lines a\Nbe an AC.
Dialogue: 0,0:20:40.41,0:20:42.12,Default,,0000,0000,0000,,Indeed, must be perpendicular.