# How can vectors be represented?

Mar 2, 2015

A vector $\vec{v}$ can be represented as a pointed arrow drawn in space:

The length of the arrow (relative to some kind of reference or scale) represents the relative magnitude of the vector while the arrow head gives us the direction in which the vector operates.

Another way to represent a vector is by giving its components .
Basically, in space, you choose three unit vectors ( of magnitude $1$ and directions the positive ones of the $x , y \mathmr{and} z$ axes) indicated as $\vec{i} , \vec{j} \mathmr{and} \vec{k}$ and then you do a linear combination of these three unit vectors using three components (=numbers such as $5 , 4 \mathmr{and} 6$).

For example, $\vec{u} = 5 \vec{i} + 4 \vec{j} + 6 \vec{k}$

Basically you can think of it as a recipe of a cake where $\vec{i} , \vec{j} \mathmr{and} \vec{k}$ are the ingredients:
To "make" vector $\vec{u}$ you need to mix (=combine linearly):
5 units of ingredient $\vec{i}$;
4 units of ingredient $\vec{j}$;
6 units of ingredient $\vec{k}$.