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.

〈4, 0〉, 〈2, 2〉

Accepted Answer

Welcome back. Everyone in this problem. We want to find the angle between the vectors 15 0 and 53. And we want to express our angle in degrees and round it to two decimal places for our answer choices. A says it's 39.06 degrees. B 59.04 degrees C 30.96 degrees and D 90.36 degrees. Now, what are we trying to figure out here? We want the anger between the two vectors. So visually what we really want here is that if we have our vectors 15 0, which would look something like this. OK. Let's say that we're just visualizing it on our X and Y axis and we have our vector 53, which should probably be something here. Then we want the anger between them. Let's say that that is anger theta. Now, how can we figure that out? Well, recall recall that the angle between two vectors is equal to the inverse cosine of their dot product divided by the product of their magnitudes. So if we call our two vectors A and B, we can figure out their dot product and their magnitudes to eventually solve for their angles. So let's let a be equal to 15 0. OK. And B equal to 53 then starting with their dot product. OK. Recall that the dot To find the dot product of two vectors means to multiply their respective components. So we're going to multiply the X components of the vector and the Y components of the vector. In this case, since we have XY for A here and Xy for B here to multiply the X components means we're going to multiply 15 by five. And to multiply the Y components means we're going to multiply zero by three. And then we're going to add them together 15, multiplied by five equals 75 and zero, multiplied by three equals zero. So their dock product of A and B is 75 next to find the magnitude of a means that we're going to find the square root of the sum of the squares of their X and Y components. In this case, that means we're going to find the square root for a of 15 squared. It's X component plus zero squared, it's Y component, no zero squared equals zero. So we're really finding the square root of 15 squared. In this case, the square cancels the square root. And that means the magnitude of A would simply be 15 units next to find the magnitude of B, we do the same thing. So we're going to find the square root of its X component five squared and the square of its Y component that would be three squared, five squared equals 25 3 squared equals nine and 25 plus nine equals 34. So the magnitude of B is the square root of 34 know that we have its dark product and the magnitudes of A and B we can solve for the anger between them because now this means that theta OK is going to be equal to the inverse cosine of 75 A dot product divided by 15 multiplied by the square root of 34. To figure out the angle, we can put this expression into our calculator. And when you do that and round it to two decimal places, you should get the to be equal to 30.96 degrees. Therefore, the angle between the vectors is 30.96 degrees. And when we look back on our answer choices that would have been answer choice. C Thanks a lot for watching everyone. I hope this video helped.

Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈4, 0〉, 〈2, 2〉

Key Concepts

Dot Product

Magnitude of a Vector

Cosine of the Angle

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