Sound good?

So that's 3,1.

What if we wanted to plot the vector?

-2, 4.

Well we go over to over -2, and up 4.

1, 2, 3, 4.

So we can plot multiple vectors in the plane as points.

You can do the same thing with 3D vectors if you're good at drawing 3D.

I am not, unfortunately.

So, I wont give you an example there.

However, when you get to four dimensional and

five dimensional vectors it becomes awfully hard to draw pictures.

Yet, all the mathematical wonders we can do with 2 and

3D vectors we can do with 5D vectors as well.

So now we're going to talk about the basic operations you can do with vectors.

And the reason we introduce them with vectors

is because we want to understand their applications to functions as well.

What is a very simple operation?

This first simple operation is just called the sum operation.

Let's say we have the vector x equals, you can put numbers, but I'm just

going to leave them as variables for now so that they can mean anything you want.

And I'm going to have them be arbitrarily dimensioned.

So, what is the dimensionality of this vector?

This is just ND vector, a list of N numbers.

How do we take the sum?

Well, it's very simple.

All you do is add all of the elements together.

Not too hard, right?

So, with the sum operation.

What goes in?

A vector goes in, that was a list of numbers.

And what comes out?

Another vector?

No, a scalar, which is just a single number.

And the reason the output is a scalar is because each of the elements

in the vector x was a scalar.

So when we add a bunch of scalars together, we get another scalar.

All right, next and most important operation is the dot product.

This will come up over and over again in your life.

And if you understand it, you will be a better person.

So how does the dot product work?

First of all, how do we write the dot product?

So let's say vector x = (x1,

x2,..., xn,

and the vector y = y1,

y2, ..., yn.

Then the dot product is just written as x.y.

Pretty crazy.

In order to compute the dot product, all we do is multiply together

the elements of x and y, element ys and then add the whole thing up.

We'll start with x1 and y1, multiply those together.

And to that we add the product of x2y2, and to that we add the product of x3y3.

And we keep going and going and

going until we get to the end, where we add the product xnyn.