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Current time:0:00Total duration:5:23

Vector magnitude from initial & terminal points

Video transcript

what we have depicted here we could call vector W and you can see from this diagram that its initial point is right over here it's the point negative seven comma positive three and it's terminal point is this point right over here which is the point two comma negative one what I want to do in this video is think about what is the magnitude of our vector and if you're saying what do I mean by magnitude well one way to think about it is what is the length of this vector how long is it pause this video and see if you can figure it out based on the information that's given well one thing that might jump out at you is that the magnitude of this vector the length of this vector is really just the distance between these two points and so if you want the magnitude you just have to apply essentially the distance formula here which is essentially just the Pythagorean theorem so we could do is construct a right triangle I will do that like this so this height in red that would be our change in Y that would be our change in Y and then what I am doing in this light blue color this would be our change in X change change in X and we know from the Pythagorean theorem that the length of the hypotenuse which would be the magnitude of our vector that that is going to be equal to that's going to be equal to the square root of our change in x squared change in x squared plus change in Y squared plus change in Y squared and so what will this be well what is our change in X our change in X our change in X you could view it as your X final minus X initial so this would be 2 minus negative 7 so this is 2 minus negative 7 which is equal to positive 9 and so this would be 9 squared and then what is our change in Y our change in Y you could view this as your Y final which is negative one minus your Y initial which is three minus three which is equal to negative four and you did indeed go down by four so this is going to be negative four and so our magnitude is going to be equal to the square root of 9 squared is 81 plus negative 4 squared is 16 and so what is that going to be let's see if you add 6 to it that gets 287 and you guys know 10 a square root of 97 so this is going to be equal to the square root of 97 which I don't think can be simplified any more but if you wanted to estimate what that is that's almost the square root of 100 so this number is going to be a little bit less than 10 is the magnitude of this vector and in this case we were able to do that from its initial point and its ending point now another way that a vector might be specified they might just be given an X component and a Y component and so for in this for example in this in this situation you could actually define our vector W by the sum of two vectors one of which is this in the blue color one of each which is the X component so you could view this as the X component of W and then the other is the Y component you could do this as you can view this as the Y component of W and you can immediately see that that Y component is the same as our change in Y and the X component is the same thing as our change in X and so sometimes you will see something like this the vector the vector W is equal to and it might look like coordinates but they're really giving you the components so the X component is positive 9 the X component is positive 9 and then the Y component is negative 4 it is negative 4 now you might say hey well what's something like this all I know is the x and y component I don't know where it exactly starts to that and that's actually on purpose because a vector you only care about the magnitude in the direction and this is actually specifying both if you wanted the magnitude here you just take the square root of the sum of the squares of the magnitudes so once again the square root of nine squared plus negative four squared is going to be the square root of 97 so you want the magnitude in the direction which this will specify but you can shift it around all that you want this vector W you could also have it starting you could also have it starting right over here and going nine in the positive x-direction and then negative four in the positive y-direction and or negative four down and so it might look something like this and so once again you can shift vectors around you care about magnitude and direction but hopefully this gives you a sense of how to find magnitude given the components or given the starting and ending points