00:01
I think the way i would tackle this one is we're going to draw the x and y axis.
00:07
Now the x and y axis is just relative to the standard basis, 1 ,0, 0 ,1.
00:15
So nothing surprising there.
00:18
So whenever we label like hash marks, 1, 2, 3, 4, 5, 6, those are all relevant to the standard basis.
00:27
1, 2, 3, 4, 5, 6.
00:33
And two, three, four, five.
00:45
And i'll be honest, i think it's much easier to compute what the coordinates are, what the linear combination of you and v would have to be to equal w.
00:56
I think it's easier to do that algebraically, but we'll try to do it with, try to do it geometrically by just looking at it.
01:06
So we're going to let you equal 1 negative 1.
01:13
We're going to let b be green.
01:20
Let's let it be blue.
01:21
Okay, v is 1 -1.
01:25
So, v we go up 1.
01:27
Relative of the standard basis, v goes up 1 and then write 1.
01:32
So this would be like one hit tick mark there.
01:37
Let me get an arrow.
01:40
And up one, right 1.
01:44
It's another length of v.
01:49
Up 1, right 1.
01:52
There's another.
01:54
Up one, right one.
01:57
Just another, up one, right one.
02:02
So here we're like drawing a uv axis essentially, up one, right one, like just going up one and right one each time.
02:10
And we're going to do the same thing with a u.
02:16
I'm going to take a second.
02:18
Let's draw the vector we want to find.
02:19
We want to find two, six.
02:22
So one, two, three, four, five, six, one, two.
02:25
We're trying to find this vector right here.
02:35
And let's do we can also go backwards with you.
02:43
But it doesn't look like we'll probably use that.
02:46
Because basically, and what you'll see in a minute is we're going to try to find a parallelogram that the green arrow w is a diagonal for that parallelogram.
02:59
And that will tell us the coordinates in uv that we need...