The unit vector notation used in Engineering mathematics is wonderfully simple and powerful.
Imagine we have a position in the x/y plane, lets call it P, and we want to form a vector from the origin to this point. We call that position vector OP.
Lets say that P is located at x=2 and y=3, now we can draw a line from the origin to P and put an arrow head along it. We have a vector.
Now lets define the unit vector (i.e. one unit in a particular direction) in the x direction as i and the unit vector in the y direction as j.
We can now say OP = 2i + 3j.
Lets try a few things ....
What is the magnitude of this vector ?
If we draw a right angled triangle from OP, we can quickly see that the magnitude of OP must be equal to the square root of (2 squared + 3 squared) = sqrt(13).
What is the unit vector of OP ?
It must be OP/sqrt(13) i.e. one unit in the direction of OP.
What about if we want to add another vector (OQ = 4i + 1j) to OP ?
OP + OQ = (4 + 2)i + (3 + 1)j = 6i + 4j.
What if I want to find the vector QO ?
QO = -OQ = -(4i + 1j) = -4i-1j
What about if we want to find the vector PQ ?
We use the head to tail rule and say PO + OQ = PQ = -(2i + 3j) + (4i + 1j) = 2i - 2j.
This approach makes the whole problem of manipulating vectors easy and more powerful than constantly referring to angles and scalar qauntities.