Question

Geometric Vectors in Cartesian Form

Answer 1

Hello there. To talk about geometric vectors in cartesian form, we have to remember some properties about linear algebra.

Given two vectors u and v, we say they are written in cartesian coordinates if they have the following notation:

`\begin{gathered} u=\langle u_1,\,u_2\rangle \\ \\ v=\langle v_1,\,v_2\rangle \end{gathered}`

Of course, this notation is for vectors in two dimensions, so we say that

`u,v\in\mathbb{V}^2`

That is the vector space with two dimensions.

We can extend this to all the plane, considering the coordinates can take all values in the real numbers, hence

`u,v\in\mathbb{R}^2`

And finally extend this to n-dimensions, but in this case we cannot understand it geometrically since we can, at most, geometrically represent a vector up to three dimensions

`u,v\in\mathbb{R}^n`

Some properties about vectors:

They are associative, that means that

`(u+v)+w=u+(v+w)`

We have also the distributive property

`(u+v)\cdot w=u\cdot w+v\cdot w`

Whereas

`\cdot\text{ is the scalar product operator}`

It also holds for cross products and other kinds of products.

They are commutative

`u+v=v+u`

This holds for the scalar product:

`u\cdot v=v\cdot u`

but it doesn't for the cross product

`u* v=-v* u`

Now, we have the geometrical view of vectors.

Say we have a point (x, y) and we want to define a vector from this point.

So we plug the tail of the vector at the origin and its tip in the point, as follows:

We can define a vector from point to point as well, but we say that they are equipollent to a vector with its tail at the origin and has the same magnitude of the vector we found.

In higher dimensions, we have

In cartesian form, we can rewrite the vectors in the following notation:

The scalar product is defined as:

`u\cdot v=\langle u_1,u_2\rangle\cdot\langle v_1,v_2\rangle=u_1v_1+u_2v_2`

For higher dimensions, it holds that

`u\cdot v=\langle u_1,u_2,\cdots,u_n\rangle\cdot\langle v_1,v_2,\cdots,v_n\rangle=\sum_(i=1)^nu_iv_i`

These are the main properties about vectors.