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.

2i + 2j, -5i - 5j

Accepted Answer

Welcome back. Everyone in this problem, we want to find the angle between the vectors 17 I plus 17 J and the negative seven I minus seven J. And we want to express our angle in degrees and round it to two decimal places for our answer choices. A says it's 90 degrees. B 180 C 180.34 degrees and D 90.54 degrees. Now, we're trying to find the anger between the two vectors and we can just quickly sketch what we think that might look like. So essentially what we're saying is that if we were to draw or X and sorry or I and J axis, then four R vector 17 I plus 17 J, which would probably be somewhere right here under the vector negative seven I minus seven J, which would be right. So in, in that quadrant, OK, in our third quadrant, then we're trying to find anger between both of those vectors, which would be the anger theta. Now, let's ask ourselves, what do we know about anger and vectors? Well, recall, OK, recall that the, between two vectors, theta two vectors A and B would be equal to the inverse cosine of their product A B and the product of their magnitudes. So if we let a be equal to one of the vectors, let's let A be equal to 17 I plus 17 J and be equals negative seven I minus, I draw that eye properly minus seven. Then we can find their dot product find their magnitudes and put it into our formula forth the angle to figure out what the anger is. So first let's start by finding their dot product. Now here for A B to find the dot product of two vectors, we multiply their components. So in this case, we would multiply the I components of A and B which are 17 and negative seven. And then we would multiply their J components which again are 17 and negative seven. And then we are going to add our two results no, 17 multiplied by negative seven equals negative 119 and the negative 119 plus negative 119 would be equal to negative 238. So that's their dot product. Next, let's find their magnitudes. Now here, the magnitude of a vector is equal to the square root of the sum of the squares of their components. So in this case, is going to be the square root or rather for a, its magnitude is going to be the square root of its I component 17 squared plus its J component, which is also 17 squared. Now notice here we are adding 17 squared twice. So it's really two multiplied by 17 squared. So we can rewrite it here as two multiplied by 17 squared. Now, based on what we know about uh roots here, we can rewrite this as the square root of two multiplied by the square root of 17 squared. Because if we recall, let me put a quick note here, recall that when working with, with, with roots, then the square root of A B equals the square root of A multiplied by the square root of B. That's another way we can write it. And the reason why that's important here is because now we can get an exact value for the magnitude of A because if that's the case, let me bring a over here for a bit. OK? Because if that's the case here, then that means the magnitude of A is going to be equal to 17 because the square root cancels the square. So it's going to be 17 multiplied by the square root of two. So that's the magnitude of A. Next, the magnitude of B would be equal to the square root of its I and J components are the sum of the squares of its I and J components, which is negative seven squared plus negative seven squared. Now here again, this is similar to what's going on in a, both. It's I and J components. Are the same. So we can rewrite it as the square root of two multiplied by the square root of negative seven all squared and no, the square cancels the square root and that leaves us with negative seven multiplied by the square root of two. Now remember we want the magnitude of B, we want it size. We're not necessarily concerned with whether it's positive or negative. So that tells us then that the magnitude of B would be equal to seven. Let me put the remind ourselves here that we want the magnitude. So the magnitude of B would be equal to seven multiplied by the square root of two. Now that we have their dark product and their magnitudes, we can substitute that into our formula to find the value of the. So this means then that theta is going to be equal to the inverse cosine of their dot products, which is negative 238 divided by 17, multiplied by the square of two multiplied by seven multiplied by the square of two. And now to figure out what that is, we can put that expression into our calculator. When we do that, we find that theta equals 180 degrees and to two decimal places, we would rate it as 180.00 degrees. So that would be the angle between both factors. If we go back to our answer choices, that would have been answer choice B 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.
2i + 2j, -5i - 5j

Key Concepts

Dot Product

Magnitude of a Vector

Angle Between Vectors

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