WEBVTT
00:00:06.624 --> 00:00:08.790
DAVID JORDAN: Hello, and
welcome back to recitation.
00:00:08.790 --> 00:00:13.740
In this problem, I'd like you to
compute the area of a triangle.
00:00:13.740 --> 00:00:15.380
This triangle sits
in space and it
00:00:15.380 --> 00:00:20.560
has its three vertices labeled
here as P_1, P_2, and P_3.
00:00:20.560 --> 00:00:22.070
So we're going to
compute this area,
00:00:22.070 --> 00:00:23.810
and we're going to do it
using the cross product, which
00:00:23.810 --> 00:00:25.029
we learned about in lecture.
00:00:25.029 --> 00:00:27.070
So why don't you take some
time to work this out,
00:00:27.070 --> 00:00:29.400
pause the video, and we'll
check back in a minute
00:00:29.400 --> 00:00:30.310
and see how I did it.
00:00:39.373 --> 00:00:41.450
Hello and welcome back.
00:00:41.450 --> 00:00:43.950
So the first thing that I
like to do with a problem
00:00:43.950 --> 00:00:47.140
like this is I like to draw a
picture so I can kind of think
00:00:47.140 --> 00:00:48.530
about what's going on.
00:00:48.530 --> 00:00:59.720
So we have this triangle
sitting out in space.
00:00:59.720 --> 00:01:02.890
And we know that we want to
take a cross product in order
00:01:02.890 --> 00:01:07.866
to compute its area, but
we need to be careful.
00:01:07.866 --> 00:01:09.365
Cross product, it
doesn't make sense
00:01:09.365 --> 00:01:10.698
to take cross product of points.
00:01:10.698 --> 00:01:13.029
What makes sense is to take
cross product of vectors.
00:01:13.029 --> 00:01:14.570
So the first thing
that we need to do
00:01:14.570 --> 00:01:17.600
is build some vectors that
describe this triangle.
00:01:17.600 --> 00:01:31.014
And the vectors that we need
to build are P_1P_2 and P_1P_3.
00:01:31.014 --> 00:01:33.430
Since we're going to use these
in a minute, let's go ahead
00:01:33.430 --> 00:01:34.960
and compute them now.
00:01:34.960 --> 00:01:44.700
So P_1P_2 is just the
difference of P_2 minus P_1.
00:01:44.700 --> 00:01:48.390
So we get a 0
minus a negative 1.
00:01:48.390 --> 00:01:53.155
So we get 1, 2, and 1.
00:01:53.155 --> 00:01:55.530
Let me just check my notes to
make sure I did that right.
00:01:55.530 --> 00:01:56.210
Good.
00:01:56.210 --> 00:02:04.030
And P_1P_3.
00:02:04.030 --> 00:02:06.500
We get again 0
minus a negative 1.
00:02:06.500 --> 00:02:11.540
So 1, we get minus
1, and then we get 1.
00:02:11.540 --> 00:02:13.120
Let me again check my notes.
00:02:13.120 --> 00:02:14.710
Very good.
00:02:14.710 --> 00:02:15.210
OK.
00:02:15.210 --> 00:02:16.800
So now that we
have these vectors,
00:02:16.800 --> 00:02:20.180
we need to remember
that if we take
00:02:20.180 --> 00:02:31.460
the absolute value of P_1P_2
cross product with P_1P_3,
00:02:31.460 --> 00:02:43.960
this will be equal to the
area of the parallelogram they
00:02:43.960 --> 00:02:44.460
enclose.
00:02:49.730 --> 00:02:52.750
So let's get started by
computing this cross product.
00:02:52.750 --> 00:02:57.690
So P_1P_2 cross P_2P_3.
00:03:01.940 --> 00:03:03.820
So, remember, to
compute a cross product,
00:03:03.820 --> 00:03:08.260
we take the determinant of a
matrix where we put in our unit
00:03:08.260 --> 00:03:11.290
normal vectors i, j, and k.
00:03:11.290 --> 00:03:14.740
And then we enter in, the
remaining entries of the matrix
00:03:14.740 --> 00:03:16.240
are just the entries
of our vectors.
00:03:16.240 --> 00:03:19.670
So we do [1, 2, 1].
00:03:19.670 --> 00:03:23.750
And [1, -1, 1].
00:03:23.750 --> 00:03:24.580
OK.
00:03:24.580 --> 00:03:27.535
And so we can compute this.
00:03:27.535 --> 00:03:34.910
And we get-- so the i component,
we get 2 minus a negative 1.
00:03:34.910 --> 00:03:37.740
So we get 3.
00:03:37.740 --> 00:03:39.190
Now the j component.
00:03:39.190 --> 00:03:43.590
If we look at the cofactor
matrix, it's just [1, 1; 1, 1],
00:03:43.590 --> 00:03:45.210
and that has determinant 0.
00:03:45.210 --> 00:03:48.310
So our middle
component is just 0.
00:03:48.310 --> 00:03:50.180
And finally the k component.
00:03:50.180 --> 00:03:53.430
We get minus 1, minus another 2.
00:03:53.430 --> 00:03:58.180
So altogether, we get minus 3.
00:03:58.180 --> 00:04:05.010
So what that tells us
now is that this quantity
00:04:05.010 --> 00:04:09.820
here, the magnitude
of the cross product,
00:04:09.820 --> 00:04:15.190
is just 3 times the
square root of 2,
00:04:15.190 --> 00:04:17.200
just looking at the length
of this vector here.
00:04:19.900 --> 00:04:22.380
So we're almost done, but
let's go back and look
00:04:22.380 --> 00:04:24.340
at what we had to start with.
00:04:24.340 --> 00:04:27.150
We were interested in
the triangle over here
00:04:27.150 --> 00:04:31.980
which was enclosed
by the vectors P_1P_2
00:04:31.980 --> 00:04:34.470
and the vectors P_1P_3.
00:04:34.470 --> 00:04:36.050
And what we just
computed is actually
00:04:36.050 --> 00:04:43.210
the area of this parallelogram,
which as you can see
00:04:43.210 --> 00:04:45.920
is twice the area
of the triangle
00:04:45.920 --> 00:04:47.530
that we're actually
interested in.
00:04:47.530 --> 00:04:55.540
So going back over here, we see
that the area of our triangle
00:04:55.540 --> 00:04:59.480
is equal to 3 root
2, and we just
00:04:59.480 --> 00:05:02.410
need to divide by 2
to get the triangle.
00:05:02.410 --> 00:05:02.910
OK?
00:05:02.910 --> 00:05:04.406
And I'll leave it at that.