3-Dimensional vectors

Notes:

3 Dimensional Vectors
3 Dimensional vectors have three components instead of two. 3D vectors are in the form

$\vec{v} = <a,b,c>$

Where
$a$ represents the $x$-coordinate
$b$ represents the $y$-coordinate
$c$ represents the $z$-coordinate

Let $\vec{v} = <a,b,c>$ and $\vec{u} = <d,e,f>$, and $c$ be a scalar.

To add two vectors, we add the corresponding components.

$\vec{v} + \vec{u} =\; <a+d,b+e,c+f>$

To subtract two vectors, we subtract the corresponding components.

$\vec{v} - \vec{u} =\; <a-d,b-e,c-f>$

To multiply a scalar with a vector, we multiple the scalar to each individual component.

$c\vec{u} = c<d,e,f>\; = \;<cd,ce,cf>$

Obtaining a Vector with 2 points.
Let the two points be $A=(x_0,y_0,z_0)$ and $B=(x_1,y_1,z_1)$. Then we can create a vector between these points by subtracting them. In other words,

$B-A = \; <x_1-x_0,y_1-y_0,z_1-z_0>$
$=\vec{BA}$

Length of a Vector
Suppose we have a vector $\vec{v} =\;<a,b,c>$. Then the length of the vector will be the formula:

$|\vec{v}|\ = \sqrt{a^2+b^2+c^2}$

Vector Properties Let $v,u,w$ be vectors and $c$ be a scalar. Here are the following vector properties:

1. $u+v=v+u$
2. $(u+v)+w=u+(v+w)$
3. $c(u+v) = cu+cv$
4. $(c+d)u=cu+du$
5. $c(du) = (cd)(u)$
6. $1u=u$

This applies to vectors in any dimension.