In the previous section, you learned how to use the arrow representation for vectors to perform vector addition and scalar multiplication.
A vector is a
mathematical expression.
Like all expressions, vectors can have lots of ‘names’—different names are better in different situations.
This section introduces the analytic representation for vectors,
and shows how vector addition and scalar multiplication are performed using this analytic form.
Consider two real numbers ‘$\,a\,,\,b\,$’ (listed in a particular order). This can convey different information, depending upon the context/interpretation:

a location (position) in a rectangular coordinate system 
specifies an arrow from $\,(0,0)\,$ to $\,(a,b)\,$ 
Analytic Representation for Vectors
horizontal and vertical components
Let $\,a\,$ and $\,b\,$ be real numbers.
The notation
$$\cssId{s25}{\langle a\,,\,b\rangle}$$
denotes the unique vector whose arrow representation (positioned at the origin)
has tail $\,(0,0)\,$ and head $\,(a,b)\,.$ The notation $\langle a\,,\,b\rangle$ is called the analytic representation of the vector. The number $\,a\,$ is called the horizontal component of the vector. The number $\,b\,$ is called the vertical component of the vector. If the arrow representing $\,\langle a\,,\,b\rangle\,$ is positioned at a point other than the origin, then the following thought process is useful:


Let $\,k\,$ be a real number (a scalar).
Let $\,\vec v = \langle a,b\rangle\,,$ where $\,a\,$ and $\,b\,$ are any real numbers.
Then:
$$
\cssId{s51}{k\vec v}
\cssId{s52}{\ =\ k\langle a,b\rangle}
\cssId{s53}{\ =\ \langle ka,kb\rangle}
$$
Notice that both components (horizontal and vertical) get multiplied by the scalar.
The horizontal component of a vector $\,\vec v\,$ is commonly named $\,v_1\,$ (and read aloud as ‘vee sub one’).
The vertical component of a vector $\,\vec v\,$ is commonly named $\,v_2\,$ (and read aloud as ‘vee sub two’).
Thus, you often see: $\,\vec v = \langle v_1,v_2\rangle\,.$
If you're conversing with someone while looking at the symbol $\,\langle v_1,v_2\rangle\,,$
then you can read it aloud as simply ‘vee one, vee two’.
They'll see the angle brackets and the subscripts, and know that you're talking about a vector.
However, if you're not looking at the symbol $\,\langle v_1,v_2\rangle\,$ (maybe you're talking over the telephone),
then for clarity it's better to read it aloud as ‘vector vee sub one, vee sub two’.
Let $\,\vec v = \langle v_1,v_2\rangle\,$ and $\,\vec w = \langle w_1,w_2\rangle\,,$
where $\,v_1\,,$ $\,v_2\,,$ $\,w_1\,,$ and $\,w_2\,$ are real numbers.
Then:
$$\begin{gather}
\cssId{s68}{\vec v + \vec w}
\cssId{s69}{\ = \ \color{blue}{\langle v_1,v_2\rangle + \langle w_1,w_2\rangle}
\cssId{s70}{\ =\ \langle v_1+w_1\,,v_2+w_2\rangle}}\cr\cr
\cssId{s71}{\vec v  \vec w}
\cssId{s72}{\ = \ \color{blue}{\langle v_1,v_2\rangle  \langle w_1,w_2\rangle}}
\cssId{s73}{\ :=\ \langle v_1,v_2\rangle + (1)\langle w_1,w_2\rangle}
\cssId{s74}{\ =\ \langle v_1,v_2\rangle + \langle w_1,w_2\rangle}
\cssId{s75}{\color{blue}{\ =\ \langle v_1w_1,v_2w_2\rangle}}
\end{gather}
$$
To add vectors, add their components.
To subtract vectors, subtract their components.
As in the previous section, it's important to observe that vector addition is both commutative and associative:
$\color{blue}{\vec v + \vec w}$  start with two arbitrary vectors 
$\ \ \ =\ \langle v_1,v_2\rangle + \langle w_1,w_2\rangle$  use the naming convention 
$\ \ \ = \langle v_1+w_1\ ,\ v_2+w_2\rangle$  vector addition 
$\ \ \ =\ \langle w_1+v_1\ ,\ w_2+v_2\rangle$  commutativity of addition of real numbers 
$\ \ \ =\ \langle w_1,w_2\rangle + \langle v_1,v_2\rangle$  vector addition (‘backwards’) 
$\ \ \ \color{blue}{=\ \vec w + \vec v}$  So, vector addition is commutative. 
$\color{blue}{(\vec u + \vec v) + \vec w}$  start with three arbitrary vectors 
$\ \ \ =\ \bigl(\langle u_1,u_2\rangle + \langle v_1,v_2\rangle\bigr) + \langle w_1,w_2\rangle$  use the naming convention 
$\ \ \ = \bigl(\langle u_1+v_1\ ,\ u_2+v_2\rangle\bigr) + \langle w_1,w_2\rangle$  vector addition 
$\ \ \ = \langle (u_1+v_1)+w_1\ ,\ (u_2+v_2)+w_2\rangle$  vector addition 
$\ \ \ = \langle u_1+(v_1+w_1)\ ,\ u_2+(v_2+w_2)\rangle$  associativity of addition of real numbers 
$\ \ \ = \langle u_1,u_2\rangle + \langle v_1+w_1\ ,\, v_2+w_2\rangle$  vector addition (‘backwards’) 
$\ \ \ = \langle u_1,u_2\rangle + \bigl(\langle v_1,v_2\rangle +\langle w_1,w_2\rangle\bigr)$  vector addition (‘backwards’) 
$\ \ \ \color{blue}{=\ \vec u + (\vec v + \vec w)}$  So, vector addition is associative. 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
