# Vectors

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The bad news today is that there will be quite a bit of math.
But the good news is that we will only do it once and it will only take something like half-hour.
There are quantities in physics which are determined uniquely by one number.
Mass is one of them.
Temperature is one of them.
Speed is one of them.
We call those scalars.
There are others where you need more than one number for instance, on a one- dimensional motion, velocity it has a certain magnitude that's the speed
but you also have to know whether it goes this way or that way.
So there has to be a direction.
Velocity is a vector and acceleration is a vector and today we're going to learn how to work with these vectors.
A vector has a length and a vector has a direction and that's why we actually represent it by an arrow.
We all have seen...
this is a vector.
Remember this
this is a vector.
If you look at the vector head-on, you see a dot.
If you look at the vector from behind, you see a cross. This is a vector and that will be our representation of vectors.
Imagine that I am standing on the table in 26.100.
This is the table and I am standing, say, at point O and I move along a straight line from O to point P so I move like so.
That's why I am on the table and that's where you will see me when you look from 26.100.
It just so happens that someone is also going to move the table
in that same amount of time
from here to there.
So that means that the table will have moved down and so my point P will have moved down exactly the same way and so you will see me now at point S.
You will see me at point S in 26.100 although I am still standing at the same location on the table.
The table has moved.
This is now the position of the table.
See, the whole table has shifted.
Now, if these two motions take place simultaneously then what you will see from where you are sitting
you will see me move in 26.100 from O straight line to S and this holds the secret behind the adding of vectors.
We say here that the vector OS
we'll put an arrow over it
is the vector OP, with an arrow over it, plus PS.
This defines how we add vectors.
There are various ways that you can add vectors.
Suppose I have here vector A and I have here vector B.
Then you can do it this way which I call the "head-tail" technique.
I take B and I bring it to the head of A.
So this is B, this is a vector and then the net result is A plus B.
This vector C equals A plus B.
That's one way of doing it.
It doesn't matter whether you take B
the tail of B to the head of A or whether you take the tail of A and bring it to the head of B.
You will get the same result.
There's another way you can do it and I call that "the parallelogram method." Here you have A.
You bring the two tails together, so here is B now so the tails are touching and now you complete this parallelogram.
And now this vector C is the same sum vector that you have here, whichever way you prefer.
You see immediately that A plus B is the same as B plus A.
There is no difference.
What is the meaning of a negative vector? Well, A minus A equals zero
vector A subtract from vector A equals zero.
So here is vector A.
So which vector do I have to add to get zero? I have to add minus A.
Well, if you use the head-tail technique
This is A.
You have to add this vector to have zero so this is minus A and so minus A is nothing but the same as A but flipped over 180 degrees.
We'll use that very often.
And that brings us to the point of subtraction of vectors.
How do we subtract vectors? So A minus B equals C.
Here we have vector A and here we have
let me write this down here
and here we have vector B.
One way to look at this is the following.
You can say A minus B is A plus minus B and we know how to add vectors and we know what minus B is.
Minus B is the same vector but flipped over so we put here minus B and so this vector now here equals A minus B.
Here's vector C, here's A minus B.
And, of course, you can do it in different ways.
You can also think of it as A plus... as C plus B is A.
Right? You can say you can bring this to the other side.
You can say C plus B is A, C plus B is A.
In other words, which vector do I have to add to B to get A? And then you have the parallelogram technique again.
There are many ways you can do it.
The head-tail technique is perhaps the easiest and the safest.
So you can add a countless number of vectors one plus the other, and the next one and you finally have the sum of five or six or seven vectors which, then, can be represented by only one.
When you add scalars, for instance, five and four then there is only one answer, that is nine.
Five plus four is nine.
Suppose you have two vectors.
You have no information on their direction but you do know that the magnitude of one is four and the magnitude of the other is five.
That's all you know.
Then the magnitude of the sum vector could be nine if they are both in the same direction
that's the maximum
or it could be one, if they are in opposite directions.