If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).

Question

Accepted Answer

Hello, today we're going to be using the given vectors to determine the scalar product. So we are told that vector A is defined as nine I plus seven J. We are told that vector B is defined as two I minus five J. And we are told that vector C is defined as eight I minus three J. We want to use these three vectors to determine the scalar product of A dot B plus C. So in order to determine the scalar product, we're first going to need to define B plus C. And in order to do that, we're going to do vector addition. So we'll be adding the vector B with the vector C. Vector B is defined as two I minus five J. And we're going to be adding the vector C which is eight I minus three J. In order to take this product, we'll be adding the I components together and the J components together. So two I plus eight, I will give us the value of 10 I and negative five J minus three J will give us the value of negative eight J. So what this means is that the vector B plus C is equal to 10, I minus eight J. So if we plug this in to the expression, we get a dotted with the vector 10, I minus eight J. Now, if we substitute in our vector A, we can rewrite the dot product as nine I plus seven J dotted with 10, I minus eight J. Now, what we wanna do is you want to take the dot product of these two vectors in order to do that, we're going to be multiplying the I components together multiply the J components together, then we'll take the sum of the two products. So the dot product of these two vectors will be defined as nine multiplied by 10 plus seven, multiplied by negative eight, nine, multiplied it by 10 will give us the value of 90 and seven multiplied by negative eight will give us the value of negative 56. Finally, 90 minus 56 will give us the value of 34. And what this means is that the dot product of A dot B plus C is going to equal to 34. And with that being said, the answer to this problem is going to be C So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).

Key Concepts

Vector Addition

Dot Product of Vectors

Component Form of Vectors

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