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 = 6i, w = 5i + 4j

Accepted Answer

Hey, everyone in this problem, we're asked to calculate the angle between the two vectors A and B and to express the angle in degrees rounded to one decimal place, we're given that A is equal to eight, I and B is equal to nine I plus six J. We have four answer choices all in degrees. Option A 48.1, option B 59.5, option C 87.9 and option D 33.7. So what we're gonna do is we're gonna go ahead and draw these vectors on a graph. OK? And we can use the typical equation dealing with the angle between two vectors. OK? We could use that the dot product is equal to the multiplication of the magnitude of a multiplied by the magnitude of B multiplied by cosine of data. OK? Work through that equation to find data. But in this case, we're gonna draw them and I'm gonna show you why we can do something a little bit simpler. So we have A, which is just eight I, OK. So that means that the X component is eight, the Y component is zero. So A is gonna lie along the X axis. OK. We have a vector A here. Now, for B, it has an X component of a nine A Y component of six, it's gonna lie somewhere up here. And we have the vector going again, pointing from the origin to this point I drew up here. And this is our vector B. Now the angle between A and B we're gonna call that beta. And in this case, because A lies along the X axis, it's just the angle between the X axis and B. Now we can draw out this triangle. We know that the height of the triangle is going to be six A because B as A Y component of six and we know that the X component of this triangle is going to be what, OK? Because again, B has an X component of nine, that triangle goes all the way to the right in line with B. So now we have a triangle where we have two sides known we want to find the angle. OK. This is a right triangle. So we can use what we know about tangent. So we're called a tinge of data. This is gonna be equal to Y divided by X. In this case, our Y value is six, our X value is nine. So 10 data is equal to six divided by nine. So data is gonna be the inverse tangent of six divided by nine. And if we work this out on a calculator, we get the data is approximately equal to 33.69 degrees. OK. So because we have this special case where one of our vectors live along the X axis, we didn't need to worry about our dot product, the magnitude of the vectors doing all those additional calculations. We just needed to look at tangent in order to calculate the angle between the two and that angle rounded to one decimal place is gonna be 33.7 degrees which corresponds with answer choice D that's it for this one. Thanks everyone for watching. I hope this video helped.

In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree. v = 6i, w = 5i + 4j

Key Concepts

Dot Product

Magnitude of a Vector

Cosine of the Angle Between Vectors

Watch next