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# Adding vectors algebraically & graphically

## Video transcript

so I have two two dimensional vectors right over here vector a and vector B and what I want to think about is what how can we define or what would be a reasonable way to define the sum of vector A plus vector V well one thing that might jump at your mind is look well each of these are two dimensional they both have two components why don't we just add the corresponding components so for the sum why don't we make the first component of the sum just the sum of the first two components of these two vectors so why don't we just make it six plus negative four well six plus negative four is equal to two and why don't we just make the second component the sum of the two second components so negative two plus four is also equal to two so we start with two two-dimensional vectors you add them together you get another two two-dimensional vectors if you think about it in terms of real coordinate space is both of these are members of r2 I'll write this down here just so we get used to the notation so vector a vector a and vector B and vector B are both members of r2 are both members of r2 which is just another way of saying that these are both two tuples they're both two-dimensional vectors right over here now this might make sense just looking at how we represented it but how does this actually make visual or conceptual sense and to do that let's actually plot these vectors let's try to represent these vectors in some way let's try to visualize them so vector a we could visualize this tells us how far this vector moves in each of these directions horizontal direction and vertical direction so if we put the I guess you could say the tail of the vector at the origin remember we don't have to put the tail of the origin but that might make it a little bit easier for us to draw it we'll go six in the horizontal direction one two three four five six and then negative two and the vertical so negative two so a vector a vector a could look like this vector a looks like that and once again the important thing is the magnitude and the direction the magnet is represented by the length of this vector and the direction is the direction that is pointed in and also just to emphasize I could have drawn vector a like that or I could have put it over here these are all these are all equivalent vectors these are all equal to vector a all I really care about is the magnitude and the direction so with that in mind let's also draw a vector B vector B in the horizontal direction goes negative four one two three four and in the vertical direction goes four one two three four so it's tail if we start at the origin if it's tail is at the origin its head would be at four it would be at negative four four so let me draw that just like that so that right over here is vector B and once again vector B we could draw it like that or we could draw it so let me copy let me paste it so this would also be another way to draw vector B this would also be another way to draw a vector B once again what I really care about is its magnitude and direction all of these green vectors have the same magnitude they held on the same length and they all have the same direction so how does the way that I've drew a vector a and B gel with the what it's sum is so let me draw its sum like this let me draw it some in this blue color and it's a blue color so the sum based on on this definition we just use a vector addition would be 2 2 so 2 2 so it would look it would look something it would look like this so how does this make sense that the sum that this purple vector plus this green vector is somehow going to be equal to this blue vector I encourage you to pause the video and think about if that even makes sense well one way to think about it is this first purple vector it shifts us this much it takes us from this point to that point and so if we were to add it let's start at this point and put the green vectors tail right there and see where it's where it ends up putting us so the green vector we've already have a version so once again we start the origin the red the vector-a takes us there now let's start over there with the green vector and see where green vector takes us and this makes sense vector A plus vector B put the tail of vector B at the head of vector a so if you if you were to start at the origin vector a takes you there then if you add on what vector B takes you it takes you right over there so relative to the origin how much did you how much did you I guess you could say shift and once again vectors don't only apply to things like displacement to comply to velocity comply to actual acceleration apply to a whole series of things but one when you visually visualize it this way you see that it does make complete sense this blue vector the sum of the two is what results where you start with vector a at that point right over there vector a takes you there then you take vector B's tail start over there and it takes you to the tip of the sum now one question you might be having is well vector A plus vector B is this but what is vector B what is vector B plus vector A plus vector a does this still work well based on the definition we had where you add the corresponding components you're still going to get the same sum vector so it should come out the same so this will just be negative 4 plus 6 is 2 4 plus negative 2 is 2 but does that make visual sense so if we start with vector B so let's say you start right over here vector B takes you right over there and then if you were to if then if you were to go there and you were start with vector a so let's do that it's actually let me let me make this a little bit let me actually let me start with a new a new vector B so let's say that that's our vector B right over there and then if actually let me give us a place where I'll have some space to work with so if let's say that's my vector B right over there and then let me get my copy of the vector a that's a good one so copy and let me paste it so I could put vector a z' tail at the tip of vector B and then it'll take me right over there so if I start here if I start if I start right over here vector B takes me there and now I'm adding to that vector a which will take me starting here will take me there and so from my original starting position I have gone I have gone this far now what is this vector well this is exactly the vector to two or another way of thinking about it this vector shifts you two in the horizontal direction and two in the vertical direction so either way you're going to get the same result and that should hopefully make visual or conceptual sense as well