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Current time:0:00Total duration:5:30

Video transcript

let's say that we have three vectors vectors a B and C and we know that vector A plus vector B plus vector B is equal to is equal to vector C now given this I have some interesting questions can you construct a scenario where the magnitude the magnitude of vector C is equal to the magnitude the magnitude of vector A plus the magnitude of vector B plus the magnitude of vector B and can you also using potentially different vectors a and B construct a scenario where the magnitude of vector C is greater than the magnitude the magnitude of vector A plus the magnitude the magnitude of vector B so I encourage you to pause this video right now and try to do that try to come up with some vectors a and B so that when you take their sum that the magnitude of the sum is equal to the sum of the magnitudes and also see if you could come up with some vectors a and B so that if you take if you take the sum of the vectors that the magnitude of the sum is actually greater than the sum of the magnitudes so see if you can come up with that well let's I'm assuming you've given a go at it and potentially you've gotten a little bit frustrated especially with the second one so the only way let's let's actually just draw some vectors so if you have vector a like this and let's say vector B vector B looks something like that then a plus B let's see we can just shift this over copy and paste a plus B is going to look a plus B is going to look like this a plus B or vector C I guess we could say is going to look is going to look like that and notice these three vectors always form a triangle and if you have a triangle one side cannot be longer than the other - then the sum of the other two sides think about it if you wanted it if you wanted this to be if you wanted this to be longer what you could try to do what you could try to do is maybe change vector B in a way so you're pushing it further and further out so you could maybe you could if you changed your vector B a little bit you can get this vector C to be longer and longer so maybe if you made your vector B like this maybe your vector B you would look something like this now your vector C is getting pretty long but it's still shorter than then the sum of these two sides and to make it equal to the sum of these two sides you essentially have to make these two vectors go in the exact same direction so to make it equal you have to have vector a looking like this and you have to change the direction of vector B or essentially construct a vector B that's going in the exact same direction so it's going in the exact same direction only in this circumstance will you get will you get this scenario where vector the magnitude of vector C is equal to so really the the largest the largest of the magnitude of the sum can be is the sum of the magnitudes and that only happens when these two are going in the same direction so these are going in the same in the exact same direction this right over here is impossible you can never get one side of a triangle being longer than the sum of the other two sides based on what we just saw now you probably say well what about the circumstance where the magnitude of our sum where the magnitude of our sum is less than is less than the magnitude of then the sum of the magnitudes I guess I could say and this is fairly difficult this is pretty much always the scenario this is what's going to always be the case when the vectors are not going to be in when the vectors are not in the same direction so if someone is someone drew a vector like this it can be drawn a little bit straighter if someone drew a vector like this and a vector like this these clearly are not going in the same direction so the sum of these two vectors the magnitude of that is going to be less than the sum of these two magnitudes so for example if the magnitude here is 5 and the magnitude here is 3 then we know that if we were to add these two things let me let me just show you so copy and paste so if we were to actually let me just cut and paste so that we can clean things up a little bit so cut and paste so let's add these two vectors so we know we know that the sum of these two which is going to be this vector right over here its magnitude its magnitude is going to be less than it's going to be less than 5 plus 3 is going to be less than 8 the only way that this magnitude could even get to 8 is if these two vectors went in the exact same went in the exact same direction