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Current time:0:00Total duration:7:42

Video transcript

let's get some practice and hopefully a little bit of intuition for adding and subtracting two-dimensional vectors so let's say that I have vector a and let's say that it's X component is I don't know 3 and it's Y component is negative 1 and let's say we also have a vector B vector B and let's say that it's X component is 2 and let's say it's Y component is 3 now let's first think about what would a plus B equal so what resulting vector would this be a plus B is equal to what and I encourage you to pause the video and think about what this would be well the convention is is if we're taking the sum of two vectors we can just add up their ex their ex components to get our new X component and add up their Y components to get the new Y component so the X component of vector A plus vector B is going to be 3 plus 2 that's going to be the X component which we know is 5 and the Y component is going to be negative 1 negative 1 plus 3 negative 1 plus three and so the resulting vector is going to have an X component of 5 and a Y component of negative 1 plus 3 is equal to 2 all right that was pretty straightforward now instead of adding them let's think about what would happen if we subtract them so let's think about what a what vector a minus vector B would be and I suspect that you might be able to guess or at least think about what would happen if you subtract vector B instead of adding vector B well as you might have guessed instead of adding the corresponding components we subtracted so our X component is going to be the X component of a minus the X component of B because it's vector a minus vector B so our X component is going to be 3 minus 2 three minus two and our Y component is going to be the Y component of a negative one minus the Y component of B so minus three minus three and so the resulting vector is going to be it's going to be the vector 3 minus 2 the X components going to be 1 and then the Y component negative 1 minus 3 is negative 4 now what I just showed you this is the convention for adding and subtracting two-dimensional vectors like vectors a and B let's think a little bit about how we can visually depict what is going on so let's the first visually depict adding vector a and vector B so let me draw some axes and so my so this could be there's my y axis Y axis let's see the highest Y value I get to is the highest Y value I get to is 3 and the lowest is negative 4 and then so let me draw the x axis someplace whoops I didn't mean to do that I did the zoom by accident all right so the x axis put it right over there there's my x axis and let's see the highest x value is 5 the lowest is 3 actually I could just focus on the I could just focus on the let me just focus on the right hand side so that's Y and then let me do X looking like that x axis and now let's see vector a is 3 comma negative 1 so 1 2 3 and negative 1 if we want to just visualize it in standard form we could put its initial point at the origin and its terminal point at the point 3 comma negative 1 and so we could draw it like that we could also shift it around as long as it has the same as long as it has the same magnitude and direction we could shift it around so that right there is our vector a now vector B 2 comma 3 we could have its initial point at the origin and just we could draw it like this so the x coordinate is 2 y coordinate 1 2 3 so it's terminal point could be there so we could just draw it like this or vector B but if we're going to add vector B to vector a like we have right over there what we want to do is shift vector B over so that it's I guess you could say tail starts at A's head or its initial point starts at a z' terminal point so let's do that so if we start right over here we're going to go to in the X direction so we're going to go 2 more in the X direction and we're going to go 3 in the Y direction so 1 2 3 so we're going to end up right over there so notice this is the same vector vector B I've just shifted it over it has the same magnitude and same direction as what I had drawn before as that vector right over there I have just shifted it down and to the right and now when we do this so this is vector a I'm adding vector B I put the tail I put the initial point of vector B at the terminal point of vector a I've shifted it over right over there so now I can figure out the resulting vector a plus B the resulting vector a plus B by going from the initial point of a to the terminal point of B so it's going to start here and then go over there and notice this vector I'm going v in the X direction and 2 in the Y direction this is the vector v comma 2 right over there that is vector A plus vector B now let's think about what happens when we do a minus B well we could still have vector a just like we drew it but now instead of putting vector B's tail or initial point at the terminal point of vector a we would want to put negative B so negative B would just be have the same magnitude but a bit in the a it would be in the opposite direction so instead of going two to the right and three up we would go two to the left and three down so this to the left and one two three down would end we would end up right about there and so if you subtract a minus B that's the same thing as a plus plus negative B so negative B is going to look something like this it's going to be the exact opposite direction it's going to look it's going to look like that and so there if you start at the initial point of a and get to the terminal point of the negative B now this is all hand drawn so it could be a little bit more precise notice we get to the point 1 comma negative 4 1 comma negative 4 so this vector right over here we do that in a different color this brown whoops that's not what I wanted to do this Brown vector right over here that is the vector a minus B and then this white one actually we do this in another color as well this magenta vector right here that is a plus B so hopefully that makes sense and if you given the components you just if you're adding the vectors add the corresponding components if you're subtracting the vectors well if you're subtracting B from a subtract B's component corresponding components from the corresponding components of a