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

Question

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

Accepted Answer

Hey, everyone in this problem, we're asked to find A dot B plus C using the following vectors 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. We're given four answer choices. A 38 B 72 C D 96. All right. So this question is asking us to take the dot product between A and B plus C. We have a question like this. We have to think about the order of operations. So we have B plus C in brackets and we need to do that addition first or we need to expand this dot product to both of those terms. OK. Now, the easiest thing to do is to work in the brackets, do the addition. First. That's gonna save us doing a couple of extra calculations and then take the dot product. So let's write out what we have. We have nine I minus four J dot I plus five J plus eight I plus three J. Now, in order to simplify this again, we're gonna simplify that second bracket where we're adding vector's B and C, we're adding vectors. So we wanna do is add the components. OK? So we have nine I minus four J. We leave that alone dotted with. Now we have I plus eight I, so I component, we're gonna have nine I and then we add the J components. We have five J plus three J which gives us plus eight J. OK? So we can just add the I components, we add the J components and we have our vector B plus C. So now we have the dot product between two vectors. OK. And we're just gonna perform this like we do any dot product, we're gonna take the first terms, multiply them together and then add the multiplication of the second two terms. OK? So let me write this out on the side. Actually, let me write it at the top if we have the vector A, which is gonna be a one I plus A two J. And we take the dot product with the vector B which we write as B one I plus B two J. Recall that we can write this as a one, B, one plus A two B two. That's exactly what we're gonna apply here. We're gonna take the I component of the first vector which is nine, multiply it by the I component of the second vector which is also nine. And then we're gonna cut, we're gonna add the J component of the first vector negative four multiplied by the J component of the second vector eight. Simplifying. We have 81 minus 32 which is gonna equal 49. And that's the result we were looking for. We had to simplify within the brackets, the vector that was added first and then perform the dot product to find that the correct answer is option C 49. Thanks everyone for watching. I hope this video helped see you in the next one.

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

Key Concepts

Vector Addition

Dot Product

Scalar Result

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