In Exercises 9–16, let
u = 2i - j, v = 3i + j, and w = i + 4j.

F...

Question

In Exercises 9–16, let
u = 2i - j, v = 3i + j, and w = i + 4j.

Find each specified scalar.
u ⋅ v + u ⋅ w

Accepted Answer

Hey, everyone in this problem, we're asked to find a dot B plus A dot C using the following vectors. We have A is equal to nine, I minus four. JB is equal to I plus five J and C is equal to eight, I plus three J. You have four answer choices. Option A 32 option B 88 option C 104 and option D 49. Now when we have A dot B plus A dot C and we have to remember the rules for what order we do these operations in. And what we need to do is do the dot products first before we add the vectors. So we're gonna take A dot B and calculate that value. We're gonna take A dot C calculate that value and then we're gonna add the two together. All right. So starting with A dot B A dot B. Well, this is gonna be equal to A one B, one plus A two B two. And recall that general formula for A dot product. We're gonna take the first component of the first vector multiply it by the first component of the second vector. And then add the second component of the first multiplied by the second component of the second vector. All right. So substituting in our values, we're gonna say that the first component is gonna be the I component. So we're gonna take the I component of A which is nine multiply it by the I component of B which is one. And then we add the J component of A negative four multiplied by the J component of B which is five. OK. Simplifying this expression, we have nine minus 20. And so we get that A dot B is equal to negative 11. So we've done our first dot product. Now we're gonna move to A dot C A dot C is gonna be, we're gonna do it the same way and we have a one C one plus A two C two, a multiplying those corresponding terms and adding up each one, the I component of A is nine multiplied by the I component of C which is A OK. And we add the J component of A is negative four multiplied by the J component of C which is three. And you have to just be careful if these are written in the opposite order. If this was written three J plus eight, I, you would still want the I terms to be multiplied together the J terms to be multiplied together. And you want those corresponding terms, not just what order they appear in simplifying, we get 72 minus 12, which gives us a value of 60. Ok. So we have our two dot products. What's left is to add them together. Ok. What we are trying to find is a dot B plus A dot C we said we needed to do those dot products first. So now we have negative 11 Pola coming from those green and blue calculations. And if we simplify and this is just equal to 49. OK. And that is that value that we were looking for. If we compare this to our answer choices, we can see that this corresponds with answer choice D. Thanks everyone for watching. I hope this video helped see you in the next one.

My Courses

Chemistry

Biology

Math

Physics

Business

Social Sciences

Programming

Product & Marketing

In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. u ⋅ v + u ⋅ w