In Exercises 17–22, find the angle between v and w. Round to the ...

Question

In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree.

v = -3i + 2j, w = 4i - j

Accepted Answer

Hey, everyone in this problem, we're asked to calculate the angle between the vectors A and B A to express it in degrees and around to one decimal place. We're given that A is equal to negative seven I plus four J and that B is equal to 14 I plus negative five J. We have four answer choices all in degrees. Option A 132.3. Option B 169.9, option C 156.4 and option D 174.9. Now we want the angle between these two vectors and let's recall our equation for dot product. OK. If we want to write A dot B, we can write this as a magnitude of a multiplied by the magnitude of B multiplied by cosine of theta or theta is the angle between A and B. OK. So this theta is what we want to calculate. Now, since we're given A and B, we can calculate the dot product and we can calculate the magnitude of each vector which is gonna allow us to solve for theta. So let's do each of these calculations separately. We're gonna start with the dot product and we're gonna do that in blue. So we have A dot B and recall that this can be written as A one B, one plus A two B two in this case, because we have two components. A where A one indicates the first element of vector A, similarly B one is the first element of vector B. And the same goes for the subscripts too. Now, the first subscript or the first element of vector A, we're gonna say it's gonna be the I component. OK. So we're gonna take the I component of A which is negative seven and multiply it by the I component of B which is 14. And then we're gonna add the J component of A four multiplied by the J component of B which is negative five, simplifying negative seven, multiplied by 14 is negative 98. And then we have plus four multiplied by negative five. So it's gonna be minus 20. A negative 98 minus 20 gives us negative 118. So we have our dot product. Now let's do the magnitude of these vectors. So we're gonna start with the magnitude of vector A, the magnitude of vector A recall is gonna be the square root of A one squared plus A two squared. OK? Because we have two components, we're just taking the square root of the sum of squares of each component. This is gonna be the square root. The first component of A is gonna be that I component. So negative seven squared plus the second component that J component which is four and we square it. This gives us that the magnitude of A is gonna be equal to the square root of 65. We're just gonna leave it like that. We're not gonna simplify yet so that we don't have any round off error in our answer. And the final thing we need to calculate before we can calculate data is this magnitude of B. We're gonna calculate that in the exact same way as A I recall, we're gonna have the square root of B one squared plus B two squared. This gives us the square root that I component of B is 14, we square it, then we add the J component which is negative five. And again, squared, if we simplify, we get the square root of 221. So now we have everything we need to substitute into our equation and solve our data. OK? If we substitute these values in, we get that negative 118 is equal to the square root of 65 multiplied by the square root of multiplied by cosine of the. And again, we want to solve for cos er for the Sorry. So let's try to isolate data. We're first gonna divide by our two square roots. We get that coast data is equal to negative divided by the square root of 65 multiplied by the square root of 221. And so data is going to be equal to the inverse cosine of all of this mess negative divided by the square root of 65 multiplied by the square root of 221. We can use our calculators now and we're gonna to approximate this to one decimal place like the question asked. And we get that data is approximately equal to 169.9 degrees. If we compare this to our answer choices, you can see that this corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree. v = -3i + 2j, w = 4i - j

Key Concepts

Dot Product

Magnitude of a Vector

Cosine of the Angle

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