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 - 4j, w = -2i - j

Accepted Answer

Hey, everyone. Welcome back in this problem. We have two vectors A and B A is equal to 11 I minus three J and B is equal to negative 10, I minus eight J. And we're asked to determine the dot products A dot B and A dot A. We have four answer choices. K. Option A has the first stop product equal to 86 and the second equals to negative 112. Option B has negative 134 and negative 112. Option C has negative 86 and 130 option D has negative 33 and 80. So let's get started. We have A dot B, OK. I call it the dot product. We're gonna take the corresponding components of each vector multiply them together and then add up all of these terms for each component. OK? Now, our vectors have two components, an I component and AJ component. So we're gonna get two terms. The first one is gonna be A one B one and we're gonna add that second term A two B two. And when we write it like this, you can imagine that our vectors are written as follows A is equal to A one I plus A two J and B is equal to B one I four B two J. OK. So the subscript one indicates the I components, the subscript two indicates the J components. But getting back to our dot product, we're gonna start with the I component. OK? These ones, so the I component of A is 11, the I component of B is negative 10, 3 of 11 multiplied by negative 10. And then we're gonna add, we're gonna do the same thing for the J component. The J component of A is negative three. The J component of B is negative A. So we add negative three multiplied by negative A. Now I can go ahead and simplify that this is equal to negative 110 plus 24 which is equal to negative 86. OK? So A dot B dot Product is equal to negative 86. Now, the only answer choice that has this answer is option C. OK? So we expect the correct answer is gonna be option C but we're gonna go ahead and calculate A dot A, the second part of this problem as well. Just to double check that we've made, made no calculation errors in the first part and that we are getting the correct answer for both. So moving to A dot A and this is very similar, but instead of having a one and B one, we just have a one and a one since A is both vectors. So we get a one multiplied by a one plus A two multiplied by A two. And we already know from the first one that A one is 11. So we have 11, multiplied by 11 and A two is negative three. So we add negative three multiplied by negative three. This gives us 121 plus nine, which is equal to 130. So we have that our dot product between A and itself is 130 which also corresponds to answer choice C and so the correct answer here is option C. That's it for this one. Thanks everyone for watching. See you in the next video.

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j

Key Concepts

Dot Product

Vector Components

Scalar Result

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