In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w

Video transcript

Hey, everyone in this problem, we're asked to find seven A minus 11 B using the following vectors. We have A is equal to negative three, I plus eight J and B is equal to eight, I minus 13 J. We have four answer choices. Option A 67 I minus 87 J. Option B 67 I plus 87 J. Option C negative 109 I plus J and option D negative 109 I minus 199 J. We're just gonna start by writing this expression we're trying to find, OK. So we have seven A minus 11 B and we're gonna substitute in the vectors. We have, this is gonna be equal to seven multiplied by the vector negative three I four. EJ Martin is 11 multiplied by the vector A I minus 13 J. Now we're called, if we're multiplying a vector by a scalar. OK. So we're multiplying A by seven. We're multiplying B by 11. We're just gonna multiply like we normally would. OK? And we want that scalar multiple to apply to each component of our vector. So what we're gonna do is take these factors out in front these scalar multiples and expand them into the brackets to apply to each term. So starting with the first, we have seven multiplied by negative three, I, that's gonna give us negative 21. I, then we have seven multiplied by AJ. OK. Expanding into the brackets. And that's gonna give us plus 56 J moving to the second term, we have negative 11 multiplied by A I. So in our expression, we have minus 88 I and then we have negative 11 multiplied by negative 13 J. That's gonna give us positive J and just be careful with the sign here, we have this negative 11. So that negative has to apply to both terms as well. OK? All right. Now we've done the expansion and it's just a matter of simplifying and collecting like terms. So if we look at our I terms, we have negative 21 I and we have negative 88 I and this is gonna give us negative 109 I doing the same for the J terms. We have plus 56 J plus 143 J which is gonna give us plus 199 J. And so the final answer A is gonna be that the expression seven A minus 11 B is equal to negative 109 I plus J which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.