To solve $450\div 3$ we are asking how many groups of 3 there are in 450. The obvious way to solve this is to subtract groups of 3 from 450, one at a time, and count each group as it is subtracted. Thus:
450
- 3
----
447
- 3
----
444
- 3
----
441
- 3
.
. (do this many many times)
.
----
6
- 3
----
3
- 3
----
0
We have to repeat this 150 times, after which the 450 has been reduced to 0. So the answer is that 450 contains 150 groups of 3.
It should be clear that there is an easy way to speed up this process: instead of removing a single group of 3 each time, we could remove 10 or even 100 groups of 3 at once. So for example:
450
- 300 (100 groups)
-----
150
- 30 (10 groups)
-----
120
- 30 (10 groups)
-----
90
- 30 (10 groups)
-----
60
- 30 (10 groups)
-----
30
- 30 (10 groups)
-----
0
Here we have removed 150 groups of 3 as before, but instead of doing them one at a time, we removed 100 right away, and then five sets of 10 groups each, for a total of 150 groups.
We can abbreviate this further:
-------
3 ) 450
We begin by removing 100 groups of 3, as before:
1
-------
3 ) 450
300 (100 groups)
---
150
Now we want to remove sets of 10 groups from 150. Removing a set of 10 groups of 3 will remove 30. Instead of removing the sets one at a time, we apply our knowledge of multiplication to see that 150 is big enough to remove 5 sets of 10 groups of 3, for a total of 150:
15
-------
3 ) 450
300 (100 groups)
---
150
150 (5 sets of 10 groups = 50 groups)
---
0
Since the total has reached 0, we do not need to remove any single groups, so we fill in the final 0 in the answer:
150
-------
3 ) 450
300 (100 groups)
---
150
150 (5 sets of 10 groups = 50 groups)
---
0
Added 2014-04-23: I learned today that my daughter is being explicitly taught this method in the fourth grade. The example she showed me was:
--------
7 ) 182
Here, she said, you might happen to know that $7\times 20=140$, so you remove the 140 from the dividend:
--------
7 ) 182
- 140 20
-----
42
Then perhaps you remember that $7\times 5 = 35$, so you remove the 35:
--------
7 ) 182
- 140 20
-----
42
- 35 5
-----
7
Then you remove the remaining 7:
--------
7 ) 182
- 140 20
-----
42
- 35 5
-----
7
- 7 1
-----
0
The remainder is now 0, so you add the right-hand column, $20+5+1$, to obtain the quotient 26.
Best Answer
Vectors should be thought of, at a first approximation, as "numbers with direction". For physical phenomena which carries a direction, such as velocity and displacement, vectors are immensely useful.
The concept of a number with direction most likely dates to antiquity, as the making of maps and sign-posts already implicitly incorporates the notion. The modern representation of a vector/point in space with an ordered triplet of numbers is often attributed to the advent of analytical geometry due to the philosopher Rene Descartes.
A different notion of vectors also arose with the "discovery" of the complex numbers by Jerome Cardan: the imaginary numbers can be thought of as living on a different direction as the real scalars (so the complex numbers form a real vector space).
Over the past 400 years or so the notion of vector gradually evolved to become what we know today, with contributions from branches of mathematics that developed into modern analysis and algebra. A nice summary of that period of development is available here. See Michael Crowe's book for a fuller description of also the Greek contributions and the influences from the 16th century in this matter.
In short, vectors shouldn't be thought of as being "invented", nor should it be attributed to one person alone.