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Current time:0:00Total duration:5:59

Video transcript

so we have right over here we have the vector U and we've defined it by giving its X&Y components it's a two dimensional vector and we have the vector W and we've graphed them the vector u its X component is 2 its Y component is negative 1 so if we put its initial point that it's at the origin the terminal point or its head will be at the point 2 comma negative 1 which is right over there and for vector W its negative 5 comma 5 so it's X its X component is negative 5 so if we start at the origin we would move 5 to the left and its Y component is positive 5 so we would move five up then to get to the head of the vector or to get to the terminal point now given these two vectors what we want to do is evaluate what 3 times the vector u plus 1/5 times the vector W is and I encourage you to pause the video and give a go at it well 3 and 1/5 are scalars they are going to scale these vectors and we're going to see them see that happen visually so we're going to scale up vector u by 3 we're going to scale down vector W we're going to multiply it by 1/5 and then we're going to add the resulting vectors so let's do that so when we scale vector u by 3 we could just view this as 3 times vector u we know is a vector 2 comma negative 1 and so that's going to be we could write it this way let me write the two let me write the negative 1 that's going to be 3 times 2 for the new X component once we scale it up and 3 times negative 1 for the new Y component and of course that's going to be result in the vector 3 times 2 our new X component is 6 and our new Y component is going to be negative 3 and so let's plot that so everything I've done just just now is that part of the expression that part of the expression so the vector 6 comma negative 3 if we started it at the origin we're going to move so let's see we're going to move one two three four five six in the X direction and negative three in the Y direction so one two three one two three so we move we get right about there so there you have it this is the vector three you we're at one two three four five six in the x-direction or six to the right and then we went down three one two three in the y-direction and notice it's in the exact same direction as vector u it is just has it just has three times the magnitude it's that's vector u that would be to you and then we get two three three times the magnitude all right now let's figure out what 1/5 times the W is so 1/5 times W well let me just write this which is going to be plus 1/5 W is the vector is the vector negative five comma five and so this is going to be plus so that's going to be let me write the components down so it's going to be plus 1/5 times 1/5 times negative five and the y component is going to be 1/5 times five and so that's going to be plus which I wrote 1/2 brains not working properly 1/5 times negative five and 1/5 times five and so this part right over here is going to be 1/5 times negative five is negative 1 and 1/5 times five is positive one and so this new vector 1/5 W is going to be W scaled down and so it's negative one comma one if we started at the origin so negative one comma one we get right over there and notice it's going in the same direction as W it's just 1/5 it's just 1/5 is long and now we just want to add these two vectors so if we add up by just looking at its components the resulting vector the resulting vector when we do and any new and a new color that I have not used yet so the resulting vector we're going to add the corresponding X components so it's going to be 6 plus negative 1 6 plus negative 1 and the resulting Y component is going to be negative 3 plus 1 negative 3 plus 1 and so the resulting vector is going to be equal to 5 comma 5 comma negative 2 and we could also see that visually if we start with this blue vector 3 times the vector U and we were to add the green vector 1/5 W well we were to add that we would just start at the head of 3 u we would when we are going to add negative 1 comma once we're going to move one to the left and one up we're going to get right over there so let me see if I can draw that so just to be a little head to tail method right over here so the head of the first vector is going to be where the tail of the next vector starts that we're adding and so the resulting vector is going to be if we started it at the tail of the first vector of 3 u right over here or at the origin and then we bring it to the head of the second vector we get once again the vector v comma negative 2 its x-component is 5 we move v to the right and we move 2 down