1 00:00:01,500 --> 00:00:05,790 The gradient of a line is a measure of how steep 2 00:00:05,790 --> 00:00:06,960 that line is. 3 00:00:11,060 --> 00:00:18,268 We may have a very steep line like 4 00:00:18,268 --> 00:00:19,169 that. 5 00:00:20,250 --> 00:00:23,166 And the gradient will be larger than a line. 6 00:00:24,590 --> 00:00:26,330 Which is a bit more shallow. 7 00:00:28,900 --> 00:00:34,624 So lines that are fairly shallow like this one will have fairly 8 00:00:34,624 --> 00:00:36,532 Lopes, fairly small gradients, 9 00:00:36,532 --> 00:00:40,600 steep lines. Have large gradients and lines. 10 00:00:41,580 --> 00:00:44,765 Which are horizontal, will have zero gradients? 11 00:00:46,030 --> 00:00:49,782 And we need to try to quantify that a little bit. Try and do 12 00:00:49,782 --> 00:00:52,194 this a little bit more mathematically so we can 13 00:00:52,194 --> 00:00:54,874 actually measure how much steeper this is than this one 14 00:00:54,874 --> 00:00:56,482 than this one, and so on. 15 00:00:57,130 --> 00:01:02,772 I'd run three line segments on this 16 00:01:02,772 --> 00:01:07,608 diagram. Let's just look at them. 17 00:01:08,950 --> 00:01:10,240 The first one. 18 00:01:10,760 --> 00:01:13,568 Is the line segment from A to D? 19 00:01:14,920 --> 00:01:17,320 Now as we move from A to D. 20 00:01:18,080 --> 00:01:21,448 The X coordinate increases from one to two. 21 00:01:23,700 --> 00:01:26,715 And the Y coordinate increases from one to five. 22 00:01:27,530 --> 00:01:33,525 Now the line segment AD is steeper than the line segment 23 00:01:33,525 --> 00:01:37,504 AC. As we move from A to C. 24 00:01:38,690 --> 00:01:43,227 Exchange is from one to two and Y changes from one to three. 25 00:01:44,560 --> 00:01:49,188 An AC in turn is steeper than the line segment AB. The line 26 00:01:49,188 --> 00:01:52,392 segment AB is in fact horizontal because as X 27 00:01:52,392 --> 00:01:56,308 increases from one to two, the Y coordinate doesn't change at 28 00:01:56,308 --> 00:01:58,088 all. It remains at one. 29 00:01:59,330 --> 00:02:04,423 Let's try and think about why mathematically, the line AD is 30 00:02:04,423 --> 00:02:06,738 steeper than the line AC. 31 00:02:07,650 --> 00:02:12,174 And the reason for this is that in both cases are X coordinate 32 00:02:12,174 --> 00:02:14,262 is changing from one to two. 33 00:02:15,210 --> 00:02:18,630 But as we move from A to D, there's a much bigger. 34 00:02:19,130 --> 00:02:24,350 Change in Y than if we move from A to see so it's this relative 35 00:02:24,350 --> 00:02:28,526 change in Y relative change in X, it's going to be important. 36 00:02:29,080 --> 00:02:34,220 What we do is we calculate the change in Y. 37 00:02:34,240 --> 00:02:40,288 Divide it by the change in X. 38 00:02:40,300 --> 00:02:46,605 That's going to be a measure of the steepness. Let's do it for 39 00:02:46,605 --> 00:02:50,000 the point AD for the points A&E. 40 00:02:50,070 --> 00:02:52,958 OK, as we move from A to D. 41 00:02:54,070 --> 00:02:58,833 Why changes from one to five? So the change in Y? 42 00:02:59,430 --> 00:03:04,185 Is 5 - 1 and the change in X while exchanges from one to two. 43 00:03:04,185 --> 00:03:07,038 So the change in X is 2 - 1. 44 00:03:07,600 --> 00:03:11,851 So this quantity, the change in Y over change in X for the 45 00:03:11,851 --> 00:03:17,083 line AD is 5 - 1, which is four 2 - 1 which is one and 46 00:03:17,083 --> 00:03:18,718 four over one is 4. 47 00:03:20,500 --> 00:03:24,801 So that's a measure of how much why changes as exchanges. 48 00:03:26,020 --> 00:03:30,417 From one to two. What about the segment AC? Let's do 49 00:03:30,417 --> 00:03:31,590 the same thing. 50 00:03:32,880 --> 00:03:36,862 While the change in Y now is from one to three. 51 00:03:37,820 --> 00:03:40,586 So the changes 3 - 1. 52 00:03:42,930 --> 00:03:49,390 The change in X well X goes from one to two, so the changes 2 - 1 53 00:03:49,390 --> 00:03:56,990 and again 3 - 1 is two. 2 - 1 is one and 2 / 1 is 2. So this 54 00:03:56,990 --> 00:04:00,410 is a measure of the relative change in X&Y. 55 00:04:01,640 --> 00:04:04,448 What about a bee? 56 00:04:04,450 --> 00:04:09,280 Well, as we move from A to B, why doesn't change at all? So 57 00:04:09,280 --> 00:04:13,765 the change in Y is 1 - 1, which is of course 0. 58 00:04:14,290 --> 00:04:19,360 And the change in X is still 2 - 1, so we get zero over 59 00:04:19,360 --> 00:04:20,712 one which is 0. 60 00:04:22,240 --> 00:04:26,660 So you see this quantity change in Y divided by changing X gives 61 00:04:26,660 --> 00:04:29,040 us a measure of the steepness of 62 00:04:29,040 --> 00:04:33,860 these lines. As we would expect, the change in Y over change in X 63 00:04:33,860 --> 00:04:39,082 for AD. Which we turned out to be 4 is greater than the change 64 00:04:39,082 --> 00:04:43,856 in Y over change in X for AC because ady is steeper than AC 65 00:04:43,856 --> 00:04:48,240 an intern. This change in Y over change in X for AC. 66 00:04:48,740 --> 00:04:53,010 Is greater than the change in Y over change in X for a bee 67 00:04:53,010 --> 00:04:55,145 because AC is steeper than a B? 68 00:04:55,730 --> 00:04:59,602 So it's this quantity which gives us the measure that we're 69 00:04:59,602 --> 00:05:03,474 looking for, and it's this quantity we define to be the 70 00:05:03,474 --> 00:05:05,234 gradient of the line segment. 71 00:05:06,650 --> 00:05:10,577 We often use the symbol M for 72 00:05:10,577 --> 00:05:17,899 gradient. So the gradient is defined to be the change 73 00:05:17,899 --> 00:05:24,828 in Y. Divided by the change in X. 74 00:05:24,830 --> 00:05:27,782 As we move from one point to a 75 00:05:27,782 --> 00:05:35,122 neighboring point. Let's do that for some general case. 76 00:05:35,122 --> 00:05:37,720 Suppose we have. 77 00:05:38,370 --> 00:05:42,048 System of coordinates 78 00:05:42,048 --> 00:05:46,220 appoint a. X1Y One. 79 00:05:48,290 --> 00:05:50,658 And a point B. 80 00:05:50,660 --> 00:05:56,780 X2Y2 And we're interested in the gradient 81 00:05:56,780 --> 00:05:59,420 of the line segment joining A&B. 82 00:06:00,510 --> 00:06:04,122 Let me put in a horizontal line 83 00:06:04,122 --> 00:06:06,908 through way. Anna vertical line. 84 00:06:07,580 --> 00:06:13,206 Through be. So there's my X axis. There's my Y Axis. 85 00:06:13,970 --> 00:06:17,225 As we move from A to B. 86 00:06:17,760 --> 00:06:20,360 Exchange is from X one. 87 00:06:20,930 --> 00:06:22,260 2X2. 88 00:06:24,230 --> 00:06:26,298 Why changes from Y1? 89 00:06:26,800 --> 00:06:27,640 To Y2. 90 00:06:29,390 --> 00:06:35,902 So the change in Y divided by the change in X. 91 00:06:36,460 --> 00:06:42,436 While the change in Y is the final value minus the initial 92 00:06:42,436 --> 00:06:46,420 value, so it's Y 2 minus Y 1. 93 00:06:46,630 --> 00:06:51,814 The change in X is X2 minus X one. 94 00:06:51,820 --> 00:06:57,449 And that is the formula that we can always use to find the 95 00:06:57,449 --> 00:07:00,047 gradient of the line joining two 96 00:07:00,047 --> 00:07:03,345 points. We can think of this 97 00:07:03,345 --> 00:07:08,960 another way. Suppose we look at this angle in here. Let's call 98 00:07:08,960 --> 00:07:10,106 that angle theater. 99 00:07:11,500 --> 00:07:14,710 Now the change in Y is just this distance here. 100 00:07:16,230 --> 00:07:18,840 Distance in there. 101 00:07:18,840 --> 00:07:23,040 And the change in X is this distance in here. 102 00:07:24,130 --> 00:07:28,225 And if we take the change in Y and divide it by the change in 103 00:07:28,225 --> 00:07:32,047 X, what we actually get is the ratio of this side of this right 104 00:07:32,047 --> 00:07:35,323 angle triangle to this side. And that's just the tangent of this 105 00:07:35,323 --> 00:07:39,478 angle here. So this quantity that we've calculated is not 106 00:07:39,478 --> 00:07:42,988 only the gradient of the line, it's also the tangent 107 00:07:42,988 --> 00:07:46,498 of the angle that the line makes with the horizontal. 108 00:07:47,760 --> 00:07:54,180 So the gradient M which we said is Y 2 minus Y, one over X2 109 00:07:54,180 --> 00:07:59,744 minus X one is also equal to the Tangent Theta, where Theta is 110 00:07:59,744 --> 00:08:03,596 the angle that the line makes with the horizontal. 111 00:08:05,570 --> 00:08:09,600 We can take this a stage further. Suppose we continue 112 00:08:09,600 --> 00:08:11,615 this line backwards until we 113 00:08:11,615 --> 00:08:13,659 meet. The X axis. 114 00:08:14,470 --> 00:08:19,198 And this angle in here between the extended line and the X 115 00:08:19,198 --> 00:08:22,466 axis. Corresponds to this angle. Here these are 116 00:08:22,466 --> 00:08:25,148 corresponding angles, so this two must also be theater. 117 00:08:26,890 --> 00:08:30,972 So In other words, the gradient of the line is also the tangent 118 00:08:30,972 --> 00:08:34,426 of the angle that the line makes with the X axis. 119 00:08:35,880 --> 00:08:42,348 Let's have a couple of examples. 120 00:08:42,940 --> 00:08:48,781 Let's choose a couple of points. Supposing a is the .34. 121 00:08:48,790 --> 00:08:50,740 And B is the point. 122 00:08:51,550 --> 00:08:58,050 814. Let's calculate the gradient of the 123 00:08:58,050 --> 00:09:01,150 line joining these two points. 124 00:09:01,160 --> 00:09:06,880 Well, the gradient is simply the difference in the Y coordinates 125 00:09:06,880 --> 00:09:08,440 14 - 4. 126 00:09:08,500 --> 00:09:10,744 Over the difference in the X 127 00:09:10,744 --> 00:09:14,368 coordinates. 8 - 3. 128 00:09:14,370 --> 00:09:21,826 14 - 4 is 10 and 8. Subtract 3 is 5 and 5 into 10 goes 129 00:09:21,826 --> 00:09:26,020 twice. So the gradient of this line is 2. 130 00:09:26,700 --> 00:09:33,509 A second other example, suppose we have the point a, which 131 00:09:33,509 --> 00:09:40,318 now has coordinates 04 and B which has coordinates 50. Let's 132 00:09:40,318 --> 00:09:42,794 do the same calculation. 133 00:09:42,810 --> 00:09:47,140 The gradient will be the difference in the Y coordinates. 134 00:09:47,140 --> 00:09:48,872 That's 0 - 4. 135 00:09:48,880 --> 00:09:54,512 Divided by the difference in the X coordinates 5 - 0. 136 00:09:54,570 --> 00:09:59,288 So this time will get minus four on the top, five. At the bottom 137 00:09:59,288 --> 00:10:00,973 we get minus four fifths. 138 00:10:02,040 --> 00:10:05,219 So this is a little bit different now because we found 139 00:10:05,219 --> 00:10:06,953 that we've got a negative number 140 00:10:06,953 --> 00:10:10,312 for our gradient. And see what that actually means. 141 00:10:10,312 --> 00:10:12,805 Let's plot the points and see what's going on. 142 00:10:14,960 --> 00:10:21,270 Point A0X coordinate Y coordinate of four. So let's put 143 00:10:21,270 --> 00:10:24,425 that there that's Point A. 144 00:10:25,230 --> 00:10:28,840 Point B has an X coordinate of five that's there. 145 00:10:29,370 --> 00:10:33,990 Y coordinate of 0, so there's my point there in there. 146 00:10:34,530 --> 00:10:37,512 And the line joining them looks 147 00:10:37,512 --> 00:10:42,205 like this. We know that this line has gradient 148 00:10:42,205 --> 00:10:43,540 minus four fifths. 149 00:10:44,810 --> 00:10:48,440 This line, as you notice, is sloping downwards as we move 150 00:10:48,440 --> 00:10:49,760 from left to right. 151 00:10:50,630 --> 00:10:54,270 And that's why the gradient turns out to be negative. 152 00:10:55,350 --> 00:10:59,646 Another way of thinking about this is that the angle that the 153 00:10:59,646 --> 00:11:04,300 line now makes with the X axis. This angle in here this theater 154 00:11:04,300 --> 00:11:06,090 is now an obtuse angle. 155 00:11:06,170 --> 00:11:10,460 Greater than 90 degrees less than 180 degrees, so we've an 156 00:11:10,460 --> 00:11:14,702 obtuse angle. A line which is sloping downwards from left to 157 00:11:14,702 --> 00:11:17,628 right. And a negative gradient. 158 00:11:18,980 --> 00:11:24,488 Let me try to summarize all that behavior. If 159 00:11:24,488 --> 00:11:26,936 you have a situation. 160 00:11:28,150 --> 00:11:30,900 Like this? 161 00:11:32,080 --> 00:11:37,075 Where the angle that the line makes with the 162 00:11:37,075 --> 00:11:38,740 horizontal is acute. 163 00:11:40,330 --> 00:11:42,370 Then the gradient. 164 00:11:42,990 --> 00:11:44,820 Will be positive. 165 00:11:45,560 --> 00:11:49,291 And the reason for that is that as you move along the line. 166 00:11:49,900 --> 00:11:53,023 As X increases, why also increases so the change 167 00:11:53,023 --> 00:11:56,840 in Y and the change in X have the same sign. 168 00:11:58,640 --> 00:12:02,000 It's also important to recognize that if we take the tangent of 169 00:12:02,000 --> 00:12:05,080 an acute angle, you get a positive number, so Tan Theater, 170 00:12:05,080 --> 00:12:08,160 which we know is the same as them, is also positive. 171 00:12:10,120 --> 00:12:16,344 What about an angle that sloping alignment sloping 172 00:12:16,344 --> 00:12:21,025 downwards? We know that the angle is now. 173 00:12:21,590 --> 00:12:23,558 Theater and it's obtuse. 174 00:12:24,700 --> 00:12:29,861 We know that the tangent of an obtuse angle is negative, and as 175 00:12:29,861 --> 00:12:31,846 we've seen, the gradient is 176 00:12:31,846 --> 00:12:37,546 negative. And that's be'cause as X increases. 177 00:12:38,300 --> 00:12:43,018 Why is decreasing so the change in Y and the change in X have 178 00:12:43,018 --> 00:12:47,062 different signs, so we take the ratio, will find out that the 179 00:12:47,062 --> 00:12:48,410 gradient is actually negative. 180 00:12:49,680 --> 00:12:50,730 And finally. 181 00:12:51,780 --> 00:12:57,045 Let's have one where theater is 0, so the angle that the line 182 00:12:57,045 --> 00:13:02,310 makes with the horizontal is 0, while tan feta is 0. So that's 183 00:13:02,310 --> 00:13:06,360 consistent with our intuition. That tells us that MSO the 184 00:13:06,360 --> 00:13:12,914 gradients 0. Let's have a look 185 00:13:12,914 --> 00:13:16,502 at some parallel 186 00:13:16,502 --> 00:13:23,416 lines. Here's 187 00:13:23,416 --> 00:13:29,132 a 188 00:13:29,132 --> 00:13:36,980 line. Let's call it L1. 189 00:13:37,940 --> 00:13:41,846 L1 will make a certain angle. 190 00:13:42,080 --> 00:13:44,288 Theater one with the X axis. 191 00:13:44,980 --> 00:13:50,381 So it's gradient, as we've seen already, is Tampa Theatre 1. 192 00:13:51,000 --> 00:13:54,490 M1, its gradient is tan. 193 00:13:55,170 --> 00:13:56,370 Theater. 194 00:13:57,420 --> 00:14:00,302 Let's put another line on this, also parallel to this first 195 00:14:00,302 --> 00:14:03,800 line. This line is L2. 196 00:14:04,620 --> 00:14:06,936 It will have a gradient M2. 197 00:14:07,620 --> 00:14:09,588 That's extend it back to the. 198 00:14:10,400 --> 00:14:14,414 Horizontal axis And let's measure this angle that would be 199 00:14:14,414 --> 00:14:19,602 theater 2. And M2 will be the tangent of Theta 2. 200 00:14:20,950 --> 00:14:25,108 Now, because these two lines are parallel. 201 00:14:26,510 --> 00:14:30,904 They cross this X axis at the same angle, Theta one and three 202 00:14:30,904 --> 00:14:34,960 to two. A corresponding angles Sophie to one must be equal to 203 00:14:34,960 --> 00:14:35,974 three to two. 204 00:14:36,590 --> 00:14:39,812 So because the to one is 3 to 2. 205 00:14:40,440 --> 00:14:45,760 10 three to one what he called 10 theater 2 so. In other words, 206 00:14:45,760 --> 00:14:46,900 M1 equals M2. 207 00:14:46,900 --> 00:14:52,910 So for two parallel lines, as you might have expected, 208 00:14:52,910 --> 00:14:56,516 intuitively, the two gradients are equal. 209 00:14:57,850 --> 00:15:01,282 And Conversely, if we have two lines for which the gradients 210 00:15:01,282 --> 00:15:05,570 are equal. Then we can deduce from that that the two lines 211 00:15:05,570 --> 00:15:12,328 must be parallel. OK, so that's parallel lines. Let's look at 212 00:15:12,328 --> 00:15:14,251 some perpendicular lines. 213 00:15:15,260 --> 00:15:18,210 See if we can do something about the gradients of 214 00:15:18,210 --> 00:15:18,800 perpendicular lines. 215 00:15:20,650 --> 00:15:23,200 Start with a point P. 216 00:15:23,790 --> 00:15:29,675 And the origin there. And let's suppose point P has coordinates 217 00:15:29,675 --> 00:15:37,165 a speed. That means that to get to pee from oh, we go a 218 00:15:37,165 --> 00:15:43,585 distance a in the X direction and be in the Y direction. 219 00:15:44,510 --> 00:15:48,098 Now what I'm going to do now is I'm going to draw a 220 00:15:48,098 --> 00:15:50,858 perpendicular line, a line that is perpendicular to Opie, and 221 00:15:50,858 --> 00:15:54,170 I'm going to do that by taking opian, rotating it through 90 222 00:15:54,170 --> 00:15:57,482 degrees. So the point P will move around here and it will 223 00:15:57,482 --> 00:15:58,862 move to appoint up there 224 00:15:58,862 --> 00:16:00,924 somewhere. And let's call that 225 00:16:00,924 --> 00:16:04,505 new point Q. I'm going to try to figure out what the 226 00:16:04,505 --> 00:16:05,240 coordinates of QR. 227 00:16:06,780 --> 00:16:12,032 This angle hit in here is 90 degrees because Opie and oq are 228 00:16:12,032 --> 00:16:17,141 perpendicular. Now to get from O to pee, wee had to go 229 00:16:17,141 --> 00:16:20,851 horizontally a distance A and vertically be. So if this 230 00:16:20,851 --> 00:16:25,303 triangle shifts around over here to get from Otak you will have 231 00:16:25,303 --> 00:16:27,529 to go vertically a distance a. 232 00:16:30,800 --> 00:16:32,505 And then horizontally a distance 233 00:16:32,505 --> 00:16:36,133 be. So you see, we've just shifted this triangle, rotated 234 00:16:36,133 --> 00:16:40,124 it round through 90 degrees, and doing that we can then read off 235 00:16:40,124 --> 00:16:41,659 the coordinates of Point Q. 236 00:16:42,360 --> 00:16:44,670 Q will have an X coordinate of 237 00:16:44,670 --> 00:16:50,880 minus B. And AY coordinate of A. 238 00:16:51,670 --> 00:16:56,700 That's now calculate the gradient of the line opi. Let's 239 00:16:56,700 --> 00:17:02,233 call that MOPMLP. Remember, is the change in Y divided by 240 00:17:02,233 --> 00:17:05,754 changing X as we move from OTP. 241 00:17:06,370 --> 00:17:12,670 As we move from outer P, the change in Y is B minus zero. 242 00:17:12,670 --> 00:17:18,166 The change in X is a minus zero. 243 00:17:18,170 --> 00:17:24,340 So the gradient of Opie is just be over A. 244 00:17:24,350 --> 00:17:31,400 What about the gradient of OQ? Let's call that MOQ. 245 00:17:33,040 --> 00:17:36,910 We want the change in Y divided by the change in X as we move 246 00:17:36,910 --> 00:17:37,942 from O to Q. 247 00:17:39,060 --> 00:17:46,230 Well, the change in Y as we move from outer Q is a subtract 0. 248 00:17:46,350 --> 00:17:51,900 And the change in X is minus B, subtract 0. 249 00:17:51,900 --> 00:17:55,977 So this time this simplifies to a over minus 250 00:17:55,977 --> 00:17:58,695 B or minus a over B. 251 00:18:00,780 --> 00:18:05,301 Now let's see what happens when we multiply these two results 252 00:18:05,301 --> 00:18:09,822 together. Let's take MOP and we're going to multiply it by 253 00:18:09,822 --> 00:18:17,200 MCU. That's be over a multiplied by minus a over B. 254 00:18:17,740 --> 00:18:22,492 And you see, when we do that, the aids cancel the beast 255 00:18:22,492 --> 00:18:24,472 cancel, and we're left with 256 00:18:24,472 --> 00:18:25,700 just. Minus one. 257 00:18:26,820 --> 00:18:31,275 This is a very important result if you have two perpendicular 258 00:18:31,275 --> 00:18:35,730 lines, then the product of their gradients is always minus one. 259 00:18:36,900 --> 00:18:40,123 And correspondingly, if you've got 2 lines and you find that 260 00:18:40,123 --> 00:18:43,346 when you multiply the gradients together, you get minus one, you 261 00:18:43,346 --> 00:18:46,276 can deduce from that that the lines must be perpendicular. 262 00:18:47,720 --> 00:18:50,912 Let's just have a look at an 263 00:18:50,912 --> 00:18:53,705 example. Let's have three 264 00:18:53,705 --> 00:18:57,098 points. Using A is the 265 00:18:57,098 --> 00:19:00,532 .12. Because the 266 00:19:00,532 --> 00:19:04,526 .34. And see is the point is 267 00:19:04,526 --> 00:19:10,520 not 3. And we'll ask ourselves, the question is AB. 268 00:19:10,520 --> 00:19:11,460 Perpendicular 269 00:19:12,530 --> 00:19:17,824 To AC. Question is a be perpendicular to AC? 270 00:19:18,480 --> 00:19:23,364 Well will will do this by calculating the gradient of the 271 00:19:23,364 --> 00:19:26,472 line from A to B. Let's call 272 00:19:26,472 --> 00:19:31,068 that MAB. And then we'll find the gradients of the line from A 273 00:19:31,068 --> 00:19:32,916 to C will call that Mac. 274 00:19:33,490 --> 00:19:37,594 So let's do this calculation. We want the gradient of the line 275 00:19:37,594 --> 00:19:41,698 from A to B. Well, that's simply the difference in the Y 276 00:19:41,698 --> 00:19:43,066 coordinates 4 - 2. 277 00:19:43,110 --> 00:19:47,200 Over the difference in the X coordinates 3 - 1. 278 00:19:47,200 --> 00:19:53,518 4 - 2 is two 3 - 1 is 2, so the gradient of a B is one. 279 00:19:54,880 --> 00:19:56,518 What about the gradient from A 280 00:19:56,518 --> 00:20:02,974 to C? Well, the difference in the Y coordinates now is 3 - 2. 281 00:20:03,000 --> 00:20:07,480 The difference in the X coordinates is 0 - 1. 282 00:20:07,810 --> 00:20:14,530 So we've got 3 - 2 is one 0 - 1 is minus one. 283 00:20:14,540 --> 00:20:16,916 So all this simplifies which is minus one. 284 00:20:18,230 --> 00:20:23,280 If we multiply the two gradients together, maybe multiplied by 285 00:20:23,280 --> 00:20:26,310 Mac, will get one times minus 286 00:20:26,310 --> 00:20:31,328 one. Which is clearly minus one, so the gradients of these two 287 00:20:31,328 --> 00:20:35,816 lines multiplied together have a result which is minus one, and 288 00:20:35,816 --> 00:20:39,896 that means that the two lines a be an AC. 289 00:20:40,410 --> 00:20:42,118 Indeed, must be perpendicular.