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Current time:0:00Total duration:6:22

Adding & subtracting vectors end-to-end

Video transcript

let's build our intuition for visually adding in subtracting vectors so let's say that I have vector a and I add that to vector V 2 vector V and the resulting vector is vector C is vector C so what could this look like visually if we assume that a b and c are two dimensional vectors well I'm just going to draw what vector a might look like so let's say that this right over here is vector a and vector B vector B since I'm adding it to vector a I'm going to put its initial point at the terminal point of vector a and then I'm going to draw vector B so let's say vector B looks like that so that is vector B so let me label these that is vector a this is vector B and I did that so that I could figure out what the sum is what vector C is going to be so that is vector B and what would that be well we would start at the initial point of a vector a and then go to the terminal point of vector B so this right over there would be the sum so that would be vector C right over there so the important realization is if I add two vectors I would put the tail of one at the head of the other and now what's neat about this if I'm adding the order doesn't matter I could have done this the other way around I could have started with vector B I could have said vector B plus vector a is equal to vector C is equal to vector C and you could see that visually it would be a slightly different visual diagram but you get to the same place so if I start with vector B let's say I start over here let's say that that's in fact you don't have to start at the origin but let's say that was the origin so I could start with vector B draw a vector B just like that and then add vector a to it so start vector a at the terminal point of vector B and then go to and then when they just draw vector a so vector so once again a vector I can shift them around as long as I'm not changing the direction or their magnitude so vector a looks like that and notice if you now go start at the initial point of B and go to the terminal point of a you still get vector C so that's why a plus B and B plus a are going to give you the same thing now what if I instead of saying a plus B I wanted to think about what a minus B is going to be so let me write that down vector a minus vector B minus vector B and let's call that vector D that is equal to vector D so once again I could start with vector a and here order matters so vector a looks something like this this is hand drawn so it's not going to be completely perfect so vector a just like that and it's one way of thinking about subtracting vector B is instead of adding vector B the way we did here we could add negative B so negative B would have the same magnitude but just the opposite direction so that's vector a vector negative B will still start it right over here but we'll go in the opposite direction so let's do that so negative B is going to look like this it is going to look something like this so that is negative B notice same magnitude exactly opposite direction we flipped it around 180 degrees and now the resulting vector is going to be d so vector D is going to look like that vector D so C is a plus B D is a minus B or you could even call this a plus a plus negative B now with that out of the way let's let's draw some diagrams and go the other way so if we can go from the diagrams to the actual equations so let's start with let me draw an interesting one so let's say that this well let's say that's vector a vector a let's say that I'll use green let's say that that is vector B and I will now use magenta and let's say that this is vector C vector C so I encourage you to pause the video and see if you can write an equation that that defines this relationship well this is interesting because they're all going in a circle right over here you have you have you could start it you could start at let's say that you started at this is your initial point you said okay a plus B well eight if you're trying to figure out what a plus B is going to be the resulting vector would go would start here and end here but vector C is going in the opposite direction but we could instead of thinking about vector C like this we could think about the the opposite of vector C which would do so instead of calling this C I could flip this around by calling it this negative C so I could flip flip this around and now the same color this would be equal to negative C notice before I just had vector C here and it started at this point and ended at this point now I just flipped it around it has the exact opposite direction same magnitude now it is negative C and this makes it easier for us to construct an equation because negative C starts at the tip at the initial point at the of or the tail of vector a and it goes to the head of vector B or the terminal point of vector B so we can now write an equation we could say vector A plus vector B plus vector B is equal to is equal to not see it's equal to the negative of vector C so hopefully you found that interesting