Let A = 4i - 2j, B = -3i + 5j, and D = A - B. (b) Draw a coordinate system and on it show vectors A, B, and D

Video transcript

Hey, everyone in this problem, we're given the vector E which is equal to two I minus J and F which is equal to negative four I plus seven J. And we're asked to calculate the vector difference G is equal to E minus F and to use a sketch to represent the three vectors. So let's start by calculating this vector difference first. So we have G which is gonna be equal to vector E minus vector F. Now we're gonna go ahead and substitute in our two vectors. So we get that G is going to be equal to two I minus J minus. And we're gonna put F in brackets that minus is gonna have to apply to every component of F. So the brackets are really important here and it's important to make sure that that minus gets distributed to both terms so that we get the correct answer. So we have negative four I plus seven J in our brackets. Now recall when we're adding or subtracting vectors, we're just gonna add and subtract the components. OK. So what we're gonna do is kind of collect like terms and simplify just like we would with a regular subtraction if we were dealing with, say X's and Ys in an equation. So we're gonna look at our eye component in our eye terms. So we have two, I minus negative four I Well, this is gonna be two I what or I, for our J terms, we have minus J and then we have minus seven J. We've just kind of rewritten, expanded these brackets, applied this negative to both terms, distributed it and grouped by like so two I plus six, I, well, that's gonna give us or sorry, two I plus four, I is gonna give us six I six. I can I get ahead of myself there. We have negative J minus seven J three at minus eight J. OK. So we have our vector G which is gonna be six I minus eight J. So we're done part one. The second part of this problem wanted us to sketch the three vectors. So let's go ahead and draw a plane to sketch it. All right. Now, in terms of the X component, the highest X component we have is six K. Remember that the X component corresponds to the I component. The highest is six. So we need to go to at least six. So let's say two, right? Each of these tick marks is gonna be two. And again, this ask for a sketch. So it doesn't have to be perfect. I wanna try to make it accurate. All right. And we're gonna do the same thing going up. All right. Now, let's start with vector E. We're gonna go ahead and draw vector E in red and vector E is gonna be two I minus J OK. We're gonna go over to down one. We're gonna plot that point and the vector is gonna point from the origin to that point. All right. So this is Victor. Next, we're gonna go to vector F. Vector F has an X component of negative four A Y component of seven. So we're gonna go over to negative four. We're gonna go up to positive seven and I put a point and our vector is gonna point from the origin to that point. That's factor F and these vectors, these should all be straight lines. OK? So they're a little wonky from being hand drawn but they are straight lines. All right. So vector F we're done with and now we're gonna move on to vector G and vector G is the one we just calculated. So we have six, I minus eight J. So we're gonna go over to six in the X direction, negative eight in the Y direction. And vector G is going to be down here. OK. Again, pointing from the origin to that point. Vector G. All right. So we calculated vector G, we drew all three vectors on the graph. So that's it for this one. Thanks everyone for watching. I hope this video helped