Prove Property 5 of vectors algebraically for the case $ n = 3 $. Then use similar triangles to give a geometric proof.
$\therefore c(a+b)=c a+c b$
okay For the human, probably want to prove property five. Which says that if we add the distributive property of vectors, so we're going to have C times A plus B. So let's say we have vectors A and then we have vector B. Well, we know that A plus B is going to be this factor here. So C times A plus B could for example, be a smaller vector. So something like, let's undo that some smaller factor that looks like this, that would be a scalar multiple of it, or it could be some larger vector that looks like this, but regardless, it's going to have that same direction. Just a potentially different magnitude. Now, let's think about the other way. What if we had C. Times those components first? So in this case we're going to have um maybe something like this and then something like this. What we see, we could rearrange this to go right here and we'd end up getting the same factor as a result.