Find the magnitude and direction angle for each vector. Round ...

Question

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈-4, 4√3〉

Accepted Answer

Welcome back everyone in this problem for the vector negative nine and the nines multiplied by the square of five. We want to calculate its magnitude and direction angle. We are expressing our answer to the nearest 10th if required for our answer choices. A says the magnitude is nine multiplied by the square root of six and 114.1 degrees. B says it's nine multiplied by the square root of six and 65.9 degrees. C says six root six and 294.1 degrees. And D says it's the square root of six and 294.1 degrees. Now, before we figure out these vectors magnitude and the direction angle, let's try to visualize what that really looks like. So here we can draw our vector on an xy plane. OK. And the vector negative nine and nine root five means that its X component is negative nine and its wire component is nine root five. OK. So the vector, the Victor, if I draw that in red, we come somewhere right here. The point where both of those coordinates intersect. Now for that vector, its magnitude is its length and its direction angle is the size of the angle going counterclockwise from the positive x axis. And for our, for our uh vector here, we can find both of those because it has horizontal and vertical components. So in other words, it's a right angled triangle and anything that happens with a right angled triangle, we can use that to help us figure out the parts of this vector. Now we know or we can recall that the magnitude of our vector, the magnitude of our vector. Let's say that magnitude which is the length of the vector L is going to be equal to the square root of the sum of the squares of its X component and Y component likewise, it's direction angle, OK? It's direction angle theta it's called a theta. Here is going to be equal to the inverse tangent of its Y component divided by its X component. Now here, like we said, we already know that its X component is negative nine and its Y component is nine multiplied by the square root of five. So we can substitute those values to solve for its magnitude and its direction angle. Let's start with its magnitude because now that means its magnitude L is going to be equal to the square root of negative nine squared plus nine multiplied by the square root of five all squared. Now here, OK, if you notice that what we know about powers, we can rewrite the product, we can rewrite a product nine multiplied by the square root of five as nine squared multiplied by the square root of five all squared. OK. I know that's pretty important because here we know that negative nine squared equals 81 and nine squared also equals 81. So we could rewrite negative nine squared or we could replace negative nine squared with nine squared because both of them will give us 81. When we square a negative number, it gives us a positive number. And here, when we do that, we realize that both of those terms have a common, have a common factor which is nine squared. So we can factor nine squared out of our expression. And now that means we're going to have left the root of nine squared. OK, multiplied by the square root of nine squared into nine squared, which is one and nine squared into nine squared, multiplied the squared of five squared, which is the squared of five squared. Now, this is a pretty important result because if we think about it, the square root of nine squared is going to be equal to nine because the square root cancels the square and the square root of the square root of sorry, the square of the square root of five would be equal to 51 plus five equals six. So we can then say that the magnitude of our vector is nine multiplied by the square root of six. If we move on to find our anger, then our anger theta is going to be equal to the inverse tangent of the Y component of our value, which is nine multiplied by the spirit of five divided by the X component, which is negative nine, which we can rewrite as our entire quotient being negative. Because when we divide a positive by a negative or result is negative. So this will be our direction angle. Let's put this into our calculator to see what we find. Now, when you do that, you should get your direction angle to be equal to 114.1 degrees. So the magnitude of our vector is nine multiplied by the spirit of six and its direction angle is 114.1 degrees to the nearest 10th. That is when we look back on our answer choices that would have been answer choice. A thanks a lot for watching everyone. I hope this video helps.

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈-4, 4√3〉

Key Concepts

Magnitude of a Vector

Direction Angle of a Vector

Quadrants in the Coordinate Plane

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