Now in life we generally have more than just one force acting on an object.
So it is very important for us to be able to find and display resultant vectors.
You know, we are going to add and subtract vectors.
We are going to combine forces from many directions on maybe an object to get it to move, or remain
stationary.
So we need to come up with an idea, or a way, of showing this addition or subtraction of
vectors both graphically... or just simply with drawing and working with the horizontal
and vertical components.
So I have a couple of vectors drawn here.
It is 'a' and 'b'.
I have only got two.
And there are two ways to draw the resultant vector.
Like say we are going to add or subtract these vectors.
One way is the parallelogram rule and one is...
I wish I had a name for it.
If there is, I don't know what it is.
End to End.
I like to call it the Triangular Method.
That is not really very good though because if you have more than two vectors you are
not going to really draw a triangle.
But at any rate, if I want to say draw...
Let's draw vector 'a' plus vector 'b'.
Well, if I am doing the Parallelogram Rule, that means I am going to start both vectors
at the same initial point.
So there is a point.
Here is, as best I can estimate it, vector 'a'.
Here is vector 'b' And the resultant is going to be drawn through the parallelogram that
I can make with those two sides.
Just the best I can here trying to get a reasonable representation of a parallelogram.
Then I am going to draw a diagonal through that parallelogram.
That diagonal is going to be my resultant vector, or vector 'a' plus vector 'b'.
Well, ok.
What if I wanted to do vector 'b' minus vector 'a'.
Well, vector 'b'...
Let's go ahead and draw that first from an initial point.
Somewhere around like that.
Vector 'a'.
It says minus vector 'a'.
Now that means that we are going to go in the opposite direction.
It is an opposite vector.
It is negative 'a' instead of positive 'a' So instead of drawing 'a' out in this direction
I am going to draw it in the opposite direction, the same length, and let this represent -a...
or the opposite of vector 'a'.
I think I did forget to do the little negative sign on the Opposite vectors I just talked
about a minute ago.
Then we are going to draw a parallelogram, draw a diagonal through that parallelogram
and that is going to be my resultant.
Now a couple of things.
I could be talking about...
Maybe I should... do a scalar multiple.
But one restriction or one thing I don't like about the parallelogram method is if you have
more than just two forces working on the same... or working together...
if you are trying to find the resultant of more than just two vectors, I find personally
using the parallelogram rule rather difficult.
How are going to draw a parallelogram when you have three or four vectors working together
in a system or on an object.
If you want to find the resultant of more than just two vectors, that is going to be
a bit of an issue.
So what if I wanted to draw the representation of... instead of b-a... vector b minus vector
a, I want to do 1/2 vector 'b' minus vector 'a'.
I want to apply a scalar multiple of 1/2.
That is going to take vector 'b' and shrink it in half.
Cut its magnitude, or its length, down by a half.
And again have a parallelogram and draw your diagonal as your resultant.
ok, I do prefer though instead of the Parallelogram method to draw the vectors End to End.
So in other words I want to make... let's see...
How about 3a+b.
Well, 3a+b is going to be... take vector 'a'.
There is no negative in front of it.
We are going to keep it in the same direction.
Just make it 3 times as long.
So here is one length of 'a', two lengths of 'a', and there is the best I can sketch
3 lengths of 'a'.
Then instead of drawing 'b' starting from the same point I am going to go End to End
and start 'b' up here.
So vector 'b' is somewhere around that length.
Now if I wanted to add a third vector, and say well there is vector 'c'.
Instead of doing just 3a+b, I want to do 3 times vector 'a' plus vector 'b' plus vector
'c'.
And with this End to End method I can just go there is 3a, there is b, and here is vector
'c'.
I am going to go back to my initial point, draw straight to my final terminal point,
and maybe try and do a straight line... but whatever.
That comes out to be this right here.
This purple vector is 3x+b+c.
Let's not forget this is a scalar multiple, and this is the resultant vector.
So there is my preferred method of drawing vectors in space.
That is using...
let's call it the End to End method as opposed to the Parallelogram Method.
Working with horizontal and vertical components.
If you want to add vectors or subtract them, all you have to do is add the horizontal components...
or subtract them... and then add or subtract the vertical components.
And really it is pretty straight forward.
So we have vector v is -2i+7j, vector w is -3i-2j.
And you can see those again with those pointy brackets but I am using the 'i' and 'j' notation
because my book does.
I really prefer the short hand version of this.
But, whatever.
Find 2W+3v.
So we are going to do a scaler multiple on w of 2.
So it is going to be 2 times -2i+7j plus 3 times vector v... Oops I just got that backwards,
didn't I. Let's try that again.
2 times vector w, so it is going to be 2 times vector w... -3i-2j.. plus 3 times vector v,
so 3 times -2i+7j.
That comes out to be... and you just distribute through the parenthesis.
So 3 times... or 2 times -3 is -6.
2 times -2 is -4.
Then plus positive 3 times -2 is -6i... and 3 times -7 gives -21j.
And this will have a resultant vector of... let's see.
We are going to add the horizontal components, so -6 and -6 is -12i... vector i.
And -4j plus 21j is 17j.
And so this is how you add or subtract vectors and do scalar multiples with simple drawings
going End to End and this is an example of combining scalar multiples and adding and
subtracting vectors in terms of their horizontal and vertical components.