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let's do some examples figuring out the magnitude of a vector if we're just given some information about it so one of the simplest cases would be well lyft is told us the actual components of the vector so if they said vector a is equal to let's say five comma negative three this means that it's X component is positive five its Y component is negative three well if we have this then the magnitude of a the magnitude of a is just going to be and this really just comes from the distance formula which just comes from the Pythagorean theorem the magnitude of a is just going to be the square root of the X component squared so let me do that in a different color so the square root of the X component squared so five squared plus the Y component squared so plus negative three squared and this is going to be equal to the square root of 25 25 plus nine plus nine which is equal to the square root of 34 which is equal to the square root of 34 and if you want to think about this visually this is very easy to do just looking at the actual components but if you want to make sense of this Y this is essentially just the Pythagorean theorem we could draw out a quick coordinate axis right over here so that's our Y axis this is our let's see I have a Y component of negative three so let's see that is our actually let me try to draw it a little bit different let me draw it like this that is our x axis and we see it's X component is positive five so one two three four five that's five there and it's Y component it's negative three so one two three and so this is negative three and so we can draw this vector with just with its initial point remember we can always shift around a vector as long as we don't change its magnitude in two direction we can start at the origin and make it go five in the x-direction and negative 3 in the y-direction and so it's terminal point will be right over there at the point 5 comma negative 3 and so the vector the vector will look like this and if we want to figure out the magnitude that's just the length of this line and what we can do is just set up a right triangle where our change our change in Y is this negative 3 right over here that is our change in Y and our change in X is this positive 5 is that positive 5 and so this is a right triangle five squared plus you could just view the absolute value of the sign just three so five squared plus three squared is going to be the hypotenuse squared come straight out of the Pythagorean theorem