In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v...

Question

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i, w = j

Accepted Answer

Hey, everyone in this problem, we have two vectors A is equal to I and B is equal to 18 J. And we're asked to determine the dot product A dot B and A dot A, we're given four answer choices A through D and each of them just contain a different combination of the answer for each of these dot products. And we're gonna come back to those as we work through the problem. So let's start with the first that we're being asked to find and that's a dot OK. We have A dot And let's recall that the dot product OK, is gonna be given by a one B, one plus A two B two. And we multiply the corresponding terms from each vector, we add them together. In this case, we have two components. We have the I component and the J component. Um And so we have two terms in our sum. Now, when we're writing this A one B, one plus A two B two, what you can think about is having our vectors written in the following form. We have A, which is equal to A one I plus A two J and B which is equal to B one I plus B two J, OK. So the subscript one corresponds with the I component. The subscript two corresponds with the J component if we look at our vectors, OK. A is just equal to I. So A one is going to be equal to one, A two is going to be equal to zero. And for B, we only have AJ component. So B two is gonna be 18. B one is going to be zero. We're getting back to our dot Product and we have a one, one multiplied by B one which is zero plus A two, which is zero, multiplied by B two, which is 18. OK? We have one multiplied by zero. That's 00, multiplied by 18. That's 00 plus zero is going to give us zero. OK? So the dot product A dot B is going to be equal to zero. And one thing I wanna note here, if we have a dot Product, that's equal to zero, that indicates that these vectors are orthogonal. OK? So if you haven't learned about orthogonal vectors yet, that's OK. You'll learn about it in the future. And if you have learned about it, there's just one thing that you want to start to try to connect as you find these dot Products. OK? It's equal to zero. That means they are thought, all right. So moving to the second part of this problem, we have a dot A is equal to. And again, it's gonna be a one, A one plus A two A two A before we had a one B one A two B two. In this case, both vectors are A. So it just a one squared plus A two squared A one is one to a one multiplied by one plus A two, which is zero, multiplied by zero. And this gives us a dot product of A with itself that is equal to one. So looking at our answer choices, OK, we found that A dot B is equal to zero. The only answer choice that has that value is option B ok. And we found that A dot A is equal to one. Ok. Answer choice A and B both have that option, but we want both of these things to be true. And so the correct answer is B thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i, w = j

Key Concepts

Dot Product of Vectors

Unit Vectors i and j

Dot Product of a Vector with Itself

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