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.

〈2, 1〉, 〈-3, 1〉

Accepted Answer

Welcome back. Everyone in this problem, we want to find the angle between the vectors 712 and negative 85. We want to also express our angle in degrees and round it to two decimal places for our answer choices. A says it's 1.74 degrees. B 88.25 degrees C 98.25 degrees and D 71.25 degrees. Now here, essentially, if we're trying to find the anger between the two vectors, what we're really saying is that if we have our two vectors A and B where will let A be equal to 712. OK? And B be equal to negative 85. And I'm just drawing this because I, I kind of like to have an idea of what it looks like before we start. So this is probably what our two vectors would look like on the xy plane. Then essentially what we're trying to do is to find the anger between them. In this case, let's call that anger theta. Now, to figure out the Anger theater, we can recall that the anger between two vectors is equal to the inverse cosine of their dot product A B divided by the product of their magnitudes, the magnitude of A multiplied by the magnitude of B. So if we let A be equal to 712 and B be equal to negative 85, then we can figure out what the anger is by finding their dot products and their magnitudes. Let's start with their dot product. Now to find the dot product of two vectors, it simply means that we multiply their components. So we are going to multiply their X components and we're going to multiply their Y components. So in this case, A B would be equal to seven multiplied by negative eight and negative sorry and 12 multiplied by five. So we multiply their components and no, we add them no seven multiplied by negative eight equals negative 56 12 multiplied by five equals 60 the negative 56 plus 60 equals four. So the dot product of A and B equals four next to find the magnitude of our vectors. That means we're going to find the square root of the sum of the squares of their components. So the magnitude of A OK. The magnitude of A is going to be equal to the square root of the square of its X component. That's seven squared plus the square of its Y component which is 12. So it's the square root of the expression seven squared plus 12 squared, no, seven squared equals 49 and the 12 squared equals 144 and 49 plus 144 equals 193. So the magnitude of A is the square root of 193. I want to keep its exact value for no, because we'd like to put the exact value in our formula to get as close to the exact answer as possible. Likewise, the magnitude of, to find the magnitude of B we do the same thing. So we'd find the square root of negative eight squared plus five squared, negative eight squared is positive 64 5 squared is 25 and 64 plus 25 equals 89. So now that we have their dark product and their magnitudes, we can go ahead and solve for theta because now that means that the is going to be equal to the inverse cosine of four divided by the square root of 193 multiplied by the square root of 89. Now, to figure out the value of the, we can just put this into our calculator. And when you do that and round it to the nearest two decimal places or round it to two decimal places, then you should get that theater is approximately equal to 88.25 degrees. If we look back on our answer choices, B is the correct answer. 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.
〈2, 1〉, 〈-3, 1〉

Key Concepts

Dot Product

Magnitude of a Vector

Cosine of the Angle

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