 It's linear algebra time and we're continuing this look into this exciting look into vectors And what we're going to look at today specifically is a very simple idea of vector addition We're just going to do vector addition And we're going to look at it in two ways one is this as far as this geometric interpretations concerned If I have two axes and I have two vectors there is a vector and there is a vector and I just want to add them And you must have seen this before Remember this Vector has an x axis value. Let's call this x sub 1 and Here and x let's make it y sub 1 and of course it was in higher dimensional space in three dimensions and hyper dimensions Just carry on with these this one will have its own Let's make that x sub 2 and it's only y sub 2 And you can well imagine when we do this we just add those two together these two components So it'll be x 1 and x plus x 2 that will be the resultant and y 1 plus y 2 What we are really doing is the following we taking this vector and We are moving it Along so that its Tail is at the head of this first one. So it's going to be this parallel movement today. Let's make it there There we go and of the similar length and that's there this value here will be x sub 1 plus x sub 2 and This value here Let's make this that'll be y sub 1 plus y sub 2. That's very easy to see and what you actually have What you actually have is this parallelogram and the resultant vector is going to be that very simple notation Now we had scalar multiplication when it comes to matrices We also get scalar multiplication of a vector if I take this vector and I multiply it by 2 The direction its angle is never going to change. It is just going to get longer So multiplied by 2 but I can multiply it by a half and I can multiply it by negative 1 I Can multiply that by negative 1. It's just going to be in this direction and Before we carry on with that, let's just think about just doing this in Vector notation. I have my one vector vector v and that is going to have components x 1 and y 1 and That is a column vector and I have vector 2 and that is going to be x sub 2 y sub 2 Those values that we see there as I said in multiple multiple multiple dimensions So if we go v1 plus v2 That is just going to be x sub 1 plus x sub 2 and y sub 1 plus y sub 2 It's very simple idea Very simple to do that as I said in higher dimensions. It just carries on these two have to have the same dimensions They have to live in the same space though If they don't live in the same space two dimensional space one dimensional space for scalar two dimensional space Three dimensional space hyperspace They've got to live in the same space and then you can simply very simply just add These vectors these vectors to each other and that just really brings me to the or why I mentioned the scalar Multiplication is I can reverse I reverse the first one here, but let's reverse the second one here by multiplying throughout by negative 1 If I do that all I'm going to do is I'm reversing the direction of this So if I want v sub 1 minus v sub 2 if I want this That to happen very easy. It is just it is just the addition of Minus 1 the scalar multiplication of So I just reverse it and then I add it so don't get confused when you see these things being drawn this way And then some convoluted thingy happening happening with with that remembering remember that this we have this binary operation between two elements of some given set this is the set of vectors in that specific space and Edition is just the Subtraction is just addition With this scalar multiplication of negative 1 with the second element there after our binary operation So very easy to do to think about vector addition vector addition Just just as far as looking at it like this and having this very simple concept in your head as to what is happening there Once you represent of course vectors in this form and this column vector form It's a very simple thing to do vector addition and then together with that vector Subjection, let's go to methamedica and see how we can accomplish this So let's have a quick look at vector addition Let me just do two vectors my one vector is going to hold as a column vector 1 comma 2 and Plus my second one. Let's make that v obvious that we're going to do Let's make it 10 and 20 is my second one and all you can see is that it's element wise addition So 1 and 10 is 11 2 and 20 is 22 Remember if I wanted to see or one way to see it as a column vector. It's just to do this matrix form Don't think we show I showed that yesterday, but just want to it's my it looks like it is a row But it's not you can clearly see that it's not a nested list And in matrix form a Single row like this is going to be seen as a column vector So remember when we had this Just our little arrow there and I just wanted to show you how easy it would be just to turn around this one So we can do that negation So that will just be negative one times So that will be minus three and if I look at that my first one is still here from zero zero to one one Zero zero to one one, but instead of zero two three We have zero two negative three and that's how you're going to do the subtraction because if I were to say Let's say one comma two and I were to subtract for instance Let's make that then the ten comma twenty You're going to see that we have one minus ten which is negative nine and two minus twenty which is negative 18 So really vector addition and vector subtract subtraction Irrespective of the dimensions that you are in as long as it's the same dimensions It's a very very easy thing to do a vector addition inside of Mathematica