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Current time:0:00Total duration:7:59

Video transcript

we've already seen that you can visually represent a vector as an arrow where the length of the arrow is the magnitude of the vector and the direction of the arrow is the direction of the vector and if we want to represent this mathematically we could just think about well starting with its starting from the tail of the vector how much does how far away is the is the head of the vector in the horizontal direction and how far away is it in the vertical direction so for example in the horizontal direction you would have to go you would have to go this distance and then in the vertical direction you would have to go you would have to go this distance let me do that in a different color you would have to go this distance right over here and so let's just say that this distance is 2 and that this distance is 3 we could represent this vector let's call this vector V we could represent vector V as an ordered or an ordered list or a 2-tuple of so we could say we move to in the horizontal direction and three in the vertical direction three in the vertical direction so you could represent it like that you could represent vector V like this where it is 2 comma 3 2 comma 3 like that what I now want to introduce you to and we could come up with other ways of representing this two tuple is another notation this really comes out of the idea of what it means to add and scale vectors and to do that we're going to define what we call unit vectors so unit unit vectors and if we're in two dimensions we define a unit vector for each of the dimensions we're operating in if we're in three dimensions we defined a unit vector for each of the three dimensions that we're operating in and so let's do that so let's define a unit vector I and the way that we denote that it is a unit vector the way that we denote it's a unit vector is instead of putting an arrow on top we put this kind of hat on top of it so the unit vector I if we wanted to write it in this notation right over year we would say it only goes one unit in the horizontal direction and it doesn't go at all in the vertical direction so it would look something like this it would look it would look something like that that is the unit vector I and then we can define another unit vector let's call that unit vector or it's typically called J J which would look which would go only in the vertical direction and not in the horizontal direction not in the horizontal direction and it goes one unit in the vertical direction so this went one unit in the horizontal now J is going to go one unit J is going to go one unit in the vertical so J just like that now any vector any two-dimensional vector we can now represent as a as a sum of scaled up versions of I and J and you say well how do we how do we do that well you could imagine vector V right here is the sum of a vector that moves purely in the horizontal direction that has length two and a vector that moves purely in the vertical direction that has length 3 so we could say that vector V vector V we do that same blue color vector V is equal to so if we want a vector that has length 2 and it moves purely in the horizontal direction well we could just scale up the unit vector I we could just multiply two times I so let's do that is equal to 2 times our unit vector I so 2i is going to be this whole thing right over here or this whole vector let me do it in this yellow color this vector right over here you could view as 2i and then to that we're going to add 3 times J so plus plus 3 times J 3 times J let me write it like this let me get the color 3 times J once again 3 times J is going to be this vector right over here and if you add this yellow vector if you add this yellow vector right over here 2 the magenta vector you're going to get noticed we're putting the tail of the magenta vector at the head of the yellow vector and if you start at the tail of the yellow vector and you go all the way to the head of the magenta vector you have now constructed vector you have now constructed vector V so vector V you could represent it as a column vector like this two three you could represent as two comma three or you could represent it as two times I with this little hat over it plus three times J with this little hat over it I is the unit vector in the horizontal direction and the positive horizontal direction if you want to go the other way you would multiply it by a negative and J is the unit vector in the vertical direction as we'll see in future videos once you go to three dimensions you'll introduce you'll introduce a K but it's very natural to translate between these two things notice two three two three and so with that lets actually do some vector operations using this notation so let's say that I define another vector let's say it is vector let's say it's vector B and vector B is equal to let's come up with some numbers here vector B is equal to negative one times I times the unit vector I plus plus four times the unit vector in the horizontal direction so given these two vector definitions what would the vector what would be the vector V plus B be equal to encourage you to pause the video and think about it well once again we just literally have to add corresponding components we could say okay well let's let's think about what we're doing in the horizontal direction we're going to in the horizontal direction here now we're going negative one so our horizontal component is going to be two two plus negative 1 2 plus negative 1 in the horizontal direction and we're going to multiply that times the unit vector I and this once again just goes back to adding the corresponding components of the vector and then we're going to have plus 4 plus 4 or plus 3 plus 4 let me write it that a plus three plus four three plus four times the unit vector J in the vertical direction and so that's going to give us that's going to give us two I'll do a salt in this one color two plus negative one is one I and we could literally write that just as I actually let's do that let's just write that as I but we got that from two plus negative 1 is 1 1 times the vector is just going to be that vector plus + 3 + 4 7 7 J and I see this is exactly how we saw how we saw vector addition in the past is that we could also represent vector B like this we could represent it like this negative 1 negative 1 4 negative 1 4 and so if you were to add V to be you add the corresponding terms so this should be equal to if we were to add corresponding terms looking at them as column vectors that is going to be equal to 2 plus negative 1 which is 1 3 plus 4 is 7 so this is the exact same representation as this this is using unit vector notation and this is representing it as a column vector