I'll call b1 here in the vector two one,

and I could have another vector here

b2 as the vector minus two four,

and I've defined it in terms of the coordinates

e. I can then describe r in terms of,

your using those vectors b1 and b2,

is just the numbers in r would be different.

So we call the vectors we use to define the space,

these guys e or these guys b,

we call them basis vectors.

So the numbers I've used to define r,

only have any meaning when I

know about the basis vectors.

So r refer to these basis vectors e is three four,

but r referred to the basis vectors b also exists.

We start out with the numbers earlier.

So this should be amazing,

the vector r has some existence

in a deep mathematical sets,

completely independently of the coordinate system

we use to describe the numbers in the list,

describing r. All the vector

that takes us from there from the origin to that,

still exist, independently of

the numbers we used in r. This is neat.

Right. So the fundamentally idea.

Now, if the new basis vectors, these guys b,

are at 90 degrees to each other,

then it turns out that projection products

has a nice application.

We can use the projection or dot-product to find out

the numbers for r in the new basis b,

so long as we know what the bs are in

terms of e. So here I've described b1 as being two one,

as being e1 plus e2,

twice e1 plus e2.

I've described b2 as being minus two e1s plus four e2s.

If I know b in terms of e,

I'm going to be able to do,

use the projection product to

find r described in terms of the bs.

But this is a big if,

the b1 and b2 have to be at 90 degrees to each other.

If they're not we end up being in big trouble and need

matrices to do what's

called a transformation of axis from

the e to the b based on basis vectors.

We'll look at matrices later,

but this will help us out a lot for now.

Using dot-products in this special case where

the new basis vectors are orthogonal to each other,

is computation a lot faster and easier,

it's just less generic.

But if you can arrange the new axis

to be orthogonal, you should,

because it makes the calculations much

faster and easier to do.

So you can see that if I project r down onto b1,

so I look down from here,

and project down at 90 degrees,

I get a length here for scalar product,

and that's scalar projection is the shadow of r to b1.

A number of the scalar projection describes

how much of this factor I need.

The vector projection is going

to actually give me a vector in the direction of b1,

of length equal to that projection.

Now, if I take the vector projection of

r onto b2 going this way,

I'm going to get a vector in the direction of

b2 of length equal to that projection.

If I do a vector sum of that vector projection,

plus this guy's vector projection,

I'll just get r. So if I

can do those two vector projections,

and add that their vector sum,

I'll then have our b

being the numbers in those two vector projections.

So I found how to get from r

in the e set of basis vectors,

to the b set of basis vectors.

Now, how do I check that

these two new basis vectors

are at 90 degrees to each other?

I just take the dot-product.

So we said before the dot-product cos Theta was

equal to the dot of two vectors together,

so b1 and b2,

divided by their lengths.