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Video transcript

I have some example problems here from our equivalent vectors exercise on Khan Academy so let's let's go through these and like always pause the video and see if you could work through them on your own so this first one it says our vectors U and vector W equivalent and so we can see vector u here in blue and vector W right over here and we have to remember a vector is defined by both having a magnitude and a direction and so for two vectors to be equivalent they have to have the same magnitude and the same direction so when I look at these two they are clearly pointing in different directions vector U is pointing to the bottom right that's the direction it's pointing in and vector W is pointing to the bottom left so they definitely aren't they definitely are not equivalent so me so they are not equal and scratch that out so they have different different directions different directions and I let's think about the magnitude if I just look at the length of the arrows just eyeballing it they look pretty close let me let me verify that so if I'm starting at the initial point for vector u how much do I move in the X direction on the X direction I go from let's see I go from negative eight to negative three so I could say my change so my change in X is positive five my x increases by five as I go from the initial point from the x coordinate of the initial point to the x coordinate of the terminal point so that length the magnitude of just the X court the X component is five and let's see what happens in the Y direction so in the Y direction I start at y equals negative two right over here and then I go down to y equals negative eight so my change in Y change in Y is equal to is equal to negative six negative six now let's think about this one over here what's my change in X what's my change in X well I'm starting at x equals 8 and I am going to x equals 3 so my change in X is negative 5 all right that change in X is equal to this is my change in X change in X and then what's my change in Y well I start at y is equal to 8 and I go down to Y is equal to 2 so just like that so my change in Y is equal to negative 6 now based on the changes in X's and Y's I can figure out the magnitude of each of these vectors the magnitude of the magnitude of U so I could write it like this the magnitude of U sometimes you'll see the notation like this the magnitude of U of vector u I should say is going to be equal to we're just going to use the Pythagorean theorem here it's going to be the square root the length of this hour is just the hypotenuse of the right triangle so it's going to be the square root of 5 squared plus negative 6 squared so it's going to be the square root of 25 plus 36 which is equal to the square root of what is that it's going to be 60 square root of 61 I can make that radical a little bit smaller square root of 61 now what about what about vector W so the magnitude of vector W is I could use a double those double bars the magnitude of vector W well that's going to be the square root of negative 5 squared which is 25 plus negative 6 squared which is going to be 36 well that's going to be the square root of 61 as well so they have the same magnitude just different directions so let's see no they have the same magnitude but different directions yep that's the choice we like let's do one more of these our vectors U and W equivalent all right so these look pretty equivalent just eyeballing it looks like the pointed in the same direction they're going from the bottom left to the top right it looks like they're the same length but we can verify them once again verify that once again by looking at their x and y components so in the x-direction or thinking about how much we change in X and change in Y when we go from the initial point to the terminal point so in the x-direction for vector u we go from negative 7 to negative 4 so we increased by 3 and what do we do over here we go from x equals 2 to x equals 5 so our x component once again has a magnitude of 3 we are our change in X is a positive 3 and so our change in Y what is that going to be well we're going on this vector from y equals whoops I'm using the wrong we're going from Y is equal to 1 to Y is equal to 6 so the magnitude of the Y component we could say is 5 our is a positive 5 and then the change in Y over here also is a positive 5 we go from negative 7 to negative 2 it's also a positive 5 so notice our change in X is the same it's a positive 3 our change in Y is the same positive 5 so that lets us know that we have the same magnitude and direction and so these are equivalent these are equivalent vectors and in fact as we see as we'll see in the future you can actually denote a vector by its components we could say that vector U vector U is the vector v comma sorry the vector 3 comma 5 our change in X comma our change our change in Y and that is exactly the same as vector W so that is exactly the same as vector W has the same change in X same changes Y as we go from the initial point to the tip of the arrow the terminal point