So I have a vector $a =( 2 ,2 )$ and a vector $b =( 0, 1 )$.
As my teacher told me, $ab = (-2, -1 )$.
$ab = b-a = ( 0, 1 ) - ( 2, 2 ) = ( 0-2, 1-2 ) = ( -2, -1 )$
$ab = a-b = ( 2 ,2 ) - ( 0 ,1 ) = ( 2-0,2-1 ) = ( 2 ,1 )$
Seems like its the same but the negative signs are gone.
Why do I have to subtract b from a to get ab? Why not a-b or a+b?
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$\begingroup$They are related by the fact that $$\mathbf a- \mathbf b = -(\mathbf b- \mathbf a)$$
The difference is the direction. Generally, the vector from a starting point to an ending point is $$(\textrm{terminal point})-(\textrm{initial point})$$
$\endgroup$ $\begingroup$The correct thing is $\;b-a\;$ for the direction vector $\;\vec{ab}\;$. The substraction $\;a-b\;$ gives the opposite direction vector, namely $\;\vec {ba}\;$
$\endgroup$ $\begingroup$( Vector AB ) = ( Vector B ) - ( Vector A )
Think of this logically when you have the equation 10 - 2 you get 8 ( a positve value ) However if you do 2 - 10 you get the same magnitude 8 but opposite direction -8.
Use this to understand the vectors since the point of Vector AB is moving from A to B you want to know whether its moving in the positive of negative direction.
If B had a greater value of position than A then obviously it moved in the positive direction to achieve that greater value and that is why B - A (larger - smaller ) has to be positive. If it had a lesser value than A then it moved in the negative direction and that is why the value of B - A would then be negative (smaller - larger).
And in reality you are subtraction the vectors OB - OA ( O being the point of origin ) so the difference between these to vectors is a displacement.
I really hope i was of any help to answer your question.
:)
$\endgroup$ $\begingroup$I know I am late to answer. Also I do not have a lot of knowledge of Mathematics. But I will try and simplify. Please do not refrain from commenting if I get it horribly or event slightly wrong.
Now to simplify lets us first imagine things in one dimensions. That is a number line. Now we have 2 points on a number line a and b.
b can either be in the positive direction ie right of a, or in the negative direction ie left of a.
So to find the direction from a to b ie ab, we will substract a from b ie b - a. So if b is greater than a (ie in positive direction) we will get a positive answer or if b is lesser than a (ie in negative direction) we will get a negative answer.
Now when we move to 2D the same logic is applied to find the direction, hence
ab = b - a ie direction from a to b.
and ba = a - b ie direction from b to a.
Also it is important to understand that direction from a to b and b to a are different values.
Hope this helps.
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