Basis vectors are best thought of in the context of roads.
Imagine you are in a city – X which has only roads that are perpendicular to one another.
You can reach any part of the city but the only constraint is that you need to move along these perpendicular roads to get there.
Now lets say you go to another city-Y which has a different structure of roads.
In this case as well you can get from one part of the city to any other, but you have to travel these ‘Sheared cubic’ pathways to get there.
Just like these roads determine how you move about in the city, Basis Vectors encode information on how you move about on a plane. What do I mean by that ?
The basis vector of City-X is given as:
This to be read as – ” If you would like to move in City-X you can only do so by taking 1 step in the x-direction or 1 step in the y-direction ”
The basis vector of City-Y is given as:
This to be read as – ” If you would like to move in City-Y you can only do so by taking 1 step in the x-direction or 1 step along the diagonal OB ”
By having the knowledge about the Basis Vectors of any city, you can travel to any destination by merely scaling these basis vectors.
As an example, lets say need to get to the point (3,2), then in City-X, you would take 2 steps in the x-direction and 3 steps in the y-direction
And similarly in City-Y, you would take 1 step along the x -direction and 2 steps along the diagonal OB.
Destination Arrived 😀