Find the angle between each pair of vectors. Round to two deci...

Question

Find the angle between each pair of vectors. Round to two decimal places as necessary.

3i + 4j, j

Accepted Answer

Welcome back. Everyone in this problem. We want to find the angle between the vectors 12 I plus 13 J and seven J. And we want to express the angle in degrees and round it to two decimal places. For our answer choices. A says that, that it's 42.71 degrees B 24.71 degrees C 71.42 degrees and D 47.29 degrees. Now, if we're trying to find the angle between the two vectors, let's first try to get an idea of what that looks like. So let's draw it or let's try to sketch it on a pair of axes or I and J axis. OK. Now, here we're saying that we have two vectors, 12 I plus 13 J probably looks something like that and we have seven J probably look something like that. OK? And what we're saying is that we're trying to find the anger between both of those vectors. We're trying to find the anger that would probably be somewhere right here. So the question is, what do we know about vectors and angles? Well, recall that if we have two vectors A and B, then the angle between both of those vectors would be equal to the inverse cosine of their product A B divided by the product of their magnitudes. So if we here, if we can set our vectors A and B find their dot products and their magnitudes, then we should be able to find the angle between them. So first, let's get A B equal to 12 I plus 13 G. OK? And let's set B to be equal to zero I, because it doesn't have an I component plus seven J first by finding their dog product to find the dot product. It means that we would multiply their respective components. Meaning we are going to multiply the I components by the I component and the J by the J component. So that means then that A dot B is going to be equal to 12 multiplied by zero. OK. Plus 13 multiplied by seven. I don't, I can't recall if I had said that earlier, but after we multiply them, we're going to add them. So that part's very important as well. So no, 12 multiplied by zero equals 0, 13, multiplied by seven equals 9191 is the dot product of A and B. Next, let's find their magnitudes not to find the magnitude of a vector. It means that we're going to find the square root of the sum of the squares of their I and J components. So for A which is 12 I plus 13 J. Then its magnitude is going to be the square root of 12 squared plus 13 squared. No, 12 squared equals 144 13 squared, 169. And the S uh 144 plus 169 equals 313. So the magnitude of A is the square root of 330. Likewise, the magnitude of B or to find the magnitude of B, we do the same thing. So we'd find the square root of its I component zero squared plus its J component seven squared, no zero squared equals zero. So really, we're finding the square root of seven squared and the square root cancels the square. So the magnitude of B is seven units. So now that we have the magnitude of A and B and their dot product, we can find the anger between them because now that means our anger theta is going to be equal to the inverse cosine of A B which is 91 divided by the square root of 313 multiplied by seven. Now, we can put this into our calculator. OK? We can put this expression into our calculator. And when we do that and around it to two decimal places, we should get our anger theater being equal to 42.71 degrees. So the two vectors, the angle between the two vectors is 42.71 degrees. If we look back on our answer choices, the correct answer is answer choice. A thanks for watching everyone. I hope this video helped.

Find the angle between each pair of vectors. Round to two decimal places as necessary.
3i + 4j, j

Key Concepts

Vector Components and Representation

Dot Product of Vectors

Calculating the Angle Between Vectors

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