C= 5/2, P = 10, the length is 5i + 10j

still, I don't understand the concept 'In order to be parallel to A, B must be a multiple of A' Could you make this part a little bit easier with easy example?

In vector mathematics, vectors have a magnitude and a direction. In general, we have

\(\displaystyle v = a \hat{r}\)

where \(\displaystyle a\) is a scalar number and \(\displaystyle \hat{r}\) is a vector that describes the direction. The convention is to use a direction vector with length 1 (which is what the little hat means above the r), but this need not always be the case.

Let's look at some examples.

If you take a piece of paper and plot the following vectors (start at the origin):

4i

-4i

4j

-4j

You'll see that they point in different directions, but they have the same length. In these cases, \(\displaystyle a\)=4 but the direction vectors change (i, -i, j, -j).

If you take a new piece of paper and plot the following vectors:

2i+j

4i+2j

6i+3j

8i+4j

10i + 5j

You'll see that they all point in the same direction (parallel), but they are successively longer and longer vectors.

For these vectors, \(\displaystyle a\) is equal to 1, 2, 3, 4 or 5 and 2i+j is describing the direction. So we can write:

\(\displaystyle v = a (2i +j)\)

If we choose to use the convention, we describe the vector instead using

\(\displaystyle v = a \left(\frac{2}{\sqrt{5}}i + \frac{1}{\sqrt{5}}j\right)\)

but I wouldn't worry about that for now.