Edexcel C4 - Simple Vectors question

Hello, it seems to me that I should easily get this but I just can't figure out it out. It seems simple but I still don't understand it.

In this diagram: http://www.mediafire.com/convkey/3976/wv756zptcp808skzg.jpg

Why is AB = OB - OA

Why is it not AB = OA + OB? What really is the difference.

I would really appreciate if you can help me understand this.


Thanks
Reda

Vector addition of A + B is "move along A, and then stick B at the end of A and move along B" - so something of this sort:


(whilst reading the next paragraph, try to ignore whatever is to the right/above the dark blue line)

So, like above - what I did was draw the line A and the line B seperately. These are the vectors OA and OB. Then, to get OA + OB. I went along the blue line till I got to the end, moved OB along till one end of the line OB was touching the end of A and moved along OB to get to OA + OB which is the dark blue line.

(okay, now you can start looking at what's above/to the right of the dark blue line)

But. You know that addition has no order (the fancy word is commutative) so OB + OA should surely be the same thing as OA + OB. And it is, it's just that we move along the green line (OB), then stick the blue line at the end of it and move along that. This gets us to precisely the same point, as you can see.

Now - one key fact is that vectors have direction, so OA and AO are different things. One is a line pointing from O to A and the other is the line pointing from A to O. From this, we can say that OA = -AO. They are precisely the opposite directions.

So, going back to your question, let's look at:



Now, if we start at A and we want to get to B (that is, AB), we need to move down the blue line and then up along the green line.

So, if we're at A, then to get to B (hence forming the vector AB, because that's what AB really means, going from A to B) we need to move along the vector AO and then stick OB to the end of AO and move along that.

So: AB = AO + OB. But we know that AO = -OA, right? Hence: AB = -OA + OB = OB - OA.

---.

I knew it was something so simple that I missed! I kept thinking we had to start it from the origin that's why I was messing up! Thank you so much mate!

Excellent explanation! I noticed this already after lots of practice for Vectors. These sorts of small things should be explained to students rather, them trying to find it out.