addition and subtraction
You can add and subtract vectors (in the same dimension).
${\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} }$
and ${\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} }$
.
Then
${\vec{a} + \vec{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}}$
Simililarly,
${\vec{a} - \vec{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} - \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}}$
simulation - PHET
scalar multiplication
${k \cdot \vec{a} = k \cdot \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} = \begin{pmatrix} ka_1 \\ ka_2 \end{pmatrix} }$
Geometrically, you are expanding or compressing the vector.
If the scalar is negative, the direction the vector is pointing in is reversed.
TODO - images
base vectors
A vector $\textbf{i}$
, where ${\textbf{i} = \vec{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} }$
is a unit vector in direction of the x-axis.
Similary,
${\textbf{j} = \vec{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} }$
${\textbf{k} = \vec{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} }$
Using addition and scalar multiplication, it is now possible to express
${\vec{a} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} }$
as ${\textbf{a} = 3\textbf{i} + \textbf{j} + 2\textbf{k}}$
.
You, will find this notation in many older books where typesetting column vectors was not possible. (see A-Level and IB)