These are just scalar quantities, and we saw that the dot product is associative with respect to scalar quantities, so we can just change the order here.

The scalar product also allows one to define an angle between two vectors formula_30In formula_31 dimensional Euclidean space, the standard scalar product (called the dot product) is given by formula_32 Cross product in 3 dimensional space The cross product of two vectors in 3 dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.

Well, this product is going to be each of these scalar terms times the column vectors of this matrix.

So right over here, I've rewritten F, and I've rewritten it as of a product of a scalar function and a vector function so the scalar part right over here Rho of x,y

It's a scalar.

They're just scalar numbers.

It's a scalar quantity.

And we also defined scalar times some matrix a to be equal to the scalar.

This is just scalar multiplication.

This is a scalar quantitiy.

These are just scalar numbers.

This is the scalar multiple.

This is the scalar x1 times the vector v1 plus the scalar x2 times the vector v2.

But we know that this product right here can be written be as each of these scalar terms in x times its corresponding column vector in A.

This is just a scalar quantity.

That's a distance, that's a scalar.

So scalar multiplication has distributive properties.

I just multiplied out this scalar.

Now, matrix scalar multiplication, very similar idea.

Five yards scalar, five yards deep vector.

We're multiplying by the scalar of x1.

We defined scalar multiplication times a transformation of x to be equal to a scalar times the transformation of x.

All of the possible scalar multiples of v are going to be the same thing as all of the scalar multiples of our unit vector, u, which is just a scalar multiple of v.

Our second condition is that the projection of a scalar multiple should be equal to a scalar multiple of the projection.

The scalar multiples, obviously, are any real number.

So we can rewrite A times x as being equal to the scalar x1 times the vector v1 plus the scalar x2.

This is a scalar quantity, this is a scalar quantity right over here. So if you want the acceleration due to gravity,

A scalar function called the electric potential can help.

Now, let's do the same thing with scalar multiplication.

These x and y here are just scalar numbers.

So I'm multiplying my vector times the scalar first.

That's our definition of scalar multiplication time's a vector.

Remember, this is just going to be some scalar.

We get x, and multiplication is associative. c is a scalar, x1 is a scalar, so we can switch them around if we want.

And then we're adding that to this guy times the scalar x2 and then we're adding that to this guy times the scalar x3.

So this is a scalar field, a line integral of a scalar field, using this curve or this path, but what happens if we take a line integral over the same scalar field, but we do it over this reverse path?

An analogy is the difference between scalar and vector arithmetic.

I'm going to define a scalar multiple of a transformation.

Now let's see if this works with a random scalar.

So, speed is measurement without direction, it's a scalar quantity.

And we saw that this is actually a scalar value.

I just multiplied one row of A times the scalar.

You might remember from physics class, a scalar is just a regular number, or if your multiplying something by a scalar, you're just scaling it up.

This is where you're not so conscientious about directon ... ... and so you use distance, which is scalar ... ... and you use rate or speed, which is scalar.

You write the two columns again, you say, ok, this product plus this product plus this product, minus this product minus this product minus that product.