Vectors can be represented through their components.
If we have a vector A, we can decompose it
into its components in the x and y-directions
by finding the vectors, one along x and one along y,
that add up to the vector A. This
is the same thing as finding the projections of the vector A
along the x and y-axes.
Here is the projection of the vector
onto the x-axis, its x-component.
And here is the projection onto the y-axis, the y-component.
This particular vector could be written
as A is equal to minus 2i hat plus minus 2j hat.
A generic vector in two dimensions
can be written as A is equal to Ax, the x-component of A,
times i hat, the unit vector along x, plus Ay,
the y-component, times j hat, the unit vector along y.
If the vector is in three dimensions,
we will also have an Az times k hat.
What if we have the vector minus 3i hat plus 2j hat?
First we find the vector minus 3 times vector i hat
and add this to the vector 2 times j hat.
We can draw this vector anywhere.
It doesn't have to start at the origin.