WEBVTT
00:00:00.920 --> 00:00:12.430
Determine whether the planes four, two, one dot 𝐫 equals eight and negative 𝑥 over five plus three 𝑦 over 10 plus 𝑧 over five equals one are parallel or perpendicular.
00:00:13.120 --> 00:00:20.120
Looking at the equations of these two planes, the first thing we’ll want to do is identify the components of vectors that are normal to each one.
00:00:20.510 --> 00:00:25.200
This first plane’s equation, we’ll call it plane one, is given to us in vector form.
00:00:25.500 --> 00:00:30.010
This means that the vector that we dot with 𝐫 is a normal vector to the plane.
00:00:30.390 --> 00:00:35.820
If we call that normal vector 𝐧 one, we can say then that it has components four, two, one.
00:00:36.410 --> 00:00:43.050
As far as the second plane’s equation, we’ll call this plane two, this is nearly given to us in what’s called general form.
00:00:43.640 --> 00:00:48.450
To write the equation that way, we would just need a zero to appear on the right instead of a one.
00:00:48.830 --> 00:00:51.950
We can accomplish this by subtracting one from both sides.
00:00:52.240 --> 00:00:59.660
But either way, we’re interested in the values by which we multiply 𝑥 and 𝑦 and 𝑧 in this equation.
00:00:59.920 --> 00:01:04.680
That’s because it’s those values that make up the components of a vector normal to this plane.
00:01:05.140 --> 00:01:06.750
We’ll call that vector 𝑛 two.
00:01:06.970 --> 00:01:11.350
And we see it has components negative one-fifth, three-tenths, and one-fifth.
00:01:12.080 --> 00:01:16.810
Now we can note that this isn’t the only vector normal or perpendicular to plane number two.
00:01:17.240 --> 00:01:23.350
In fact, we can multiply this vector by any nonzero constant 𝐶, and it would still be normal to the plane.
00:01:23.730 --> 00:01:27.930
To make 𝑛 two a bit simpler to work with, let’s let 𝐶 equal positive five.
00:01:28.340 --> 00:01:31.870
In other words, we’re multiplying all the components of this vector by five.
00:01:32.230 --> 00:01:35.930
This gives us a resulting vector of negative one, three-halves, one.
00:01:36.630 --> 00:01:43.690
Now that we know the components of vectors that are normal to both of our planes, we can start testing whether these planes are parallel or perpendicular.
00:01:44.060 --> 00:01:52.740
In general, if two planes are parallel, then that means their normal vectors, 𝐧 one and 𝐧 two, are equal to one another to within a constant value.
00:01:53.150 --> 00:02:00.120
In other words, there exists some constant, we’ve called it 𝐾, by which we can multiply one of the normal vectors so that it equals the other.
00:02:00.700 --> 00:02:04.100
If the planes are not parallel, then they may be perpendicular.
00:02:04.450 --> 00:02:08.670
The condition for that is that the dot product of 𝐧 one and 𝐧 two equals zero.
00:02:09.490 --> 00:02:14.900
So let’s apply these tests to our two given planes, starting with a test of whether they’re parallel.
00:02:15.320 --> 00:02:21.910
We can begin to conduct this test by looking at our two normal vectors and specifically analyzing their 𝑥-components.
00:02:22.190 --> 00:02:32.410
If some constant does exist by which we can multiply 𝐧 two to equal 𝐧 one, then we can solve for the value of that constant by seeing what it must be in order for this equation to be true.
00:02:32.850 --> 00:02:38.030
If it is true that four equals 𝐾 times negative one, that means 𝐾 must equal negative four.
00:02:38.440 --> 00:02:42.210
This is the only value 𝐾 can have in order for this equation to hold true.
00:02:42.520 --> 00:02:48.680
So now what we do is we look at the 𝑦- and then the 𝑧-components and see if the same 𝐾-value works for them.
00:02:49.280 --> 00:02:51.170
Let’s try the 𝑦-components first.
00:02:51.340 --> 00:02:56.350
We want to see if two is equal to 𝐾 times three-halves, where 𝐾 is equal to negative four.
00:02:56.850 --> 00:03:00.480
Negative four times three over two though equals negative six.
00:03:00.800 --> 00:03:02.380
And that is not equal to two.
00:03:02.660 --> 00:03:08.460
So therefore, this 𝐾-value can’t be used consistently across all dimensions to make our normal vectors equal.
00:03:08.900 --> 00:03:13.850
And that means there is no such constant by which we can multiply these vectors so they are equal.
00:03:14.270 --> 00:03:19.490
That tells us that our two planes, represented by the vectors 𝐧 one and 𝐧 two, are not parallel.
00:03:20.010 --> 00:03:22.160
Let’s consider then whether they’re perpendicular.
00:03:22.500 --> 00:03:28.030
If they are, the dot product of a vector normal to one plane and one normal to the other should be zero.
00:03:28.610 --> 00:03:37.870
We see this dot product equals four times negative one plus two times three-halves plus one times one or, simplifying, negative four plus three plus one.
00:03:38.300 --> 00:03:43.610
This does equal zero, which means that 𝐧 one dot 𝐧 two equal zero for our two planes.
00:03:43.780 --> 00:03:49.630
And therefore, we can say that the two planes in this example are oriented perpendicularly to one another.