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.

〈1, 6〉, 〈-1, 7〉

Accepted Answer

Welcome back. Everyone in this problem. We want to find the angle between the vectors 1113 and the negative 1114 expressing our angle in degrees and rounding it to two decimal places for our answer choices. A says that says that it's 61.1 degrees. B says it's 11.61 degrees C 87.39 degrees and D 78.39 degrees. Now, since we're trying to find the anger between the two vectors, let's try to get an idea of what that looks like. So let's draw a sketch of our vectors on our X and Y axis. Now here, what we're seeing is that we have two vectors 1 1113. OK. It's probably somewhere here and the other negative 1114, which would be over here. And we are trying to find the angle between the vectors. Let's call that anger theta. So this is what we're really looking for. So how are we going to figure it out? What do we know about the anger between vectors? Well, recall that for two vectors A and B, the anger theta between them would be equal to the inverse cosine of their dot product divided by the product of the magnitudes of both vectors. So if we can figure out the dot product and their magnitudes, then we will be able to find the anger theta between them. Now let's let a, let's say that vector A is 1113 and the Victor B it is negative 1114. Then if that's the case, that's the case. That means then that first their dot product, which if we recall the dot product or to find the dot product of two vectors, we multiply their components and then add them. That means they do product would have been 11 multiplied by negative 11. The X components, the product of the X components uh plus 13 multiplied by 14. OK. So here we've multiplied their X components and their Y components and we've added them. That's how we find the dot product no, 11 multiplied by negative 11 equals negative 121. OK. And the 13 multiplied by 14 equals 182. And the negative 121 plus 182 equals 61. So the dark product of our two vectors equals 61. Next to find the magnitude of our vectors recall that the magnitude of any vector or a two D vector rather is equal to the square root of the sum of the squares of its X and Y components. So in this case, the magnitude of a would be equal to the square root of 11 squared plus 13 squared. Now, 11 squared equals 121 13 squared equals 169 and 121 plus 169 equals 290. So the magnitude of A is the square root of 290. Likewise, the magnitude of B and the magnitude of B is going to be equal to the square root of its X component. So that's negative 11 squared plus its Y component. 14 squared, negative 11 squared equals positive 121 14 squared equals 196. And when we add both of those, we get the magnitude of B to be equal to 317 or the square root rather of 317. Now that we have the dot product of both vectors and their magnitudes, we can find the anger between them because no, that means the anger theta between them would be equal to the inverse cosine of 61 divided by the square root of 290 multiplied by the square root of 317. To figure out the value of this angle, we can put this expression into our calculator. Now when you do that and when we round it to two decimal places, you should get our anger to be equal to 78.39 degrees. That would be the angle between the two vectors. If we look back on our answer choices, D 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.
〈1, 6〉, 〈-1, 7〉

Key Concepts

Dot Product

Magnitude of a Vector

Cosine of the Angle

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