00:01
Welcome back to another cross -product problem.
00:03
We're going to try a really messy proof that a cross -b -c is equal to a dot -c -t -c times b -mines a -a -d -b times c.
00:15
I started by writing up the definition of b -cross -c, just using our regular cross -product method.
00:23
And so what we're going to focus on now is writing out in full a cross b cross c.
00:32
So we're looking at the vector a, which we'll just call ax, ay, az, cross this entire thing.
00:47
It's going to be a bit of a mess, and i'm not going to be showing all of my work, but let's do our best here.
00:56
So if you want to write this out in full, you're welcome to work through it.
01:00
But first we ignore our first column of the matrix.
01:03
We're going to look at ay times all of this.
01:12
It'll be bx, c, y, minus b, y, c, x, x, x, x, minus a, z times all of this.
01:30
And i'm going to be working that negative sign in.
01:32
So that'll be b, z, c, x, minus bx, c, z.
01:44
And all of this is multiplied by i.
01:48
Minus.
01:50
Then we'll be covering up our second column.
01:52
We'll be looking at ax times the last component again.
02:00
It'll be bx, c, y, minus b, y, cx.
02:07
And then minus a z times the first column.
02:20
It'll be b -y -c -z minus b -z -c -y -c -y, all times j.
02:33
And then lastly, when we ignore the third column, that'll be a -x times this middle one.
02:41
Again, working in that negative sign, b -z -c -s -c -x.
02:48
X minus bx c z minus a y times the first one b y c z minus b z z c y c y minus b z c y all times k now all of this is going to get really messy i know it's already pretty messy but all we're going to do is ignore the second two terms for now we're just going to focus on what happens in the first component of this vector.
03:34
And what we're going to do is we're going to rewrite it.
03:38
So let's pull out a bx, and then what's left over is we have a y -c -y -c -y, and we have a bx here, leaving us with an a z -c -z, and then let's pull out a cx, leaving us with that's going to be a -y -y -by -all -times cx, b -y b -y plus a -z b -z.
04:37
All right, and we can verify, you can double -check my work, that all this is equal to all of this.
04:45
Now we're going to do something a little strange.
04:49
We're going to add, let's make sure i'm adding the right thing here.
04:56
We're going to add a -x, b -x, c -x, and then we'll subtract a -s...