Vector A has y-component Ay = +9.60 m. A makes an angle of 32.0° counterclockwise from the +y-axis. (a) What is the x-component of A? (b) What is the magnitude of A?

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Hey, everyone in this problem, we're told that we have a vector M with a Y component, my equal to positive 8.5 m and a counterclockwise angle of 48 degrees measured from a positive X axis. We are asked two questions here. The first is for the X component of vector and the second is for the magnitude of M. OK. So we have four answer choices A through D each with a different combination of the values for the X component of M and the magnitude of M. And we're gonna come back to those as we work through this problem. So let's start by drawing out a picture of what we have going on here. So we have this vector M and it makes an angle of 48 degrees from the positive X axis measured counterclockwise. So let's just draw, I'm gonna draw in this dotted line our axis. OK. So from the positive X axis, you have a 48 degree angle counter clock, wax 48 degrees. And this diagonal line is going to be our vector. Now, the vector N has a Y component my which we know is 8.5 m and it has an X component MX, which we're trying to look for in part A. So let's start with part, we have this triangle. This is gonna be a right angle triangle. We know the angle, we know one of the sidelines. So we can use our trigonometric ratios to find the other sideline MX. OK. Now my is the opposite side of our angle MX is the adjacent side to our angle. So we wanna use the tangent of data to relate the opposite and adjacent side. OK. So we're called the tangent of the is going to be equal to the opposite side divided by the adjacent side. So in our case, we have the tangent of 48 degrees is going to be equal to 8.5 m divided by. And that now we're looking for MX, we're gonna go ahead and rearrange. We get that MX is going to be equal to 8.5 m divided by a tangent of 48 degrees, give ourselves a little bit more room and we work this out on our calculator. We get that, that X component of our vector M is going to be about 7.653 m. All right. Now, if we look at our answer choices, option A and option C have the X component of them being 9.44 m. That is not what we found. OK. So we can eliminate option A and option C option B and D both have this X component being 7.65 m, which is what we found is we could have either of those being the correct answer. We're gonna move on to part B to figure out which one it is. Now, for part B, we can do this in a couple of different ways. OK. We're looking for the magnitude of the vector N. OK. So that's the diagonal or the hypotenuse of our time. Now, we know the two sides MX and my, so we can use Pythagorean theorem to solve for M, we can also use our trigonometric ratios again, just choosing a different one to account for the different side. We're gonna go ahead and use the trig ratio. The reason I do that is just so that we can avoid some round off error. OK. If we use the value of MX, we just calculated there's gonna be a tiny bit of round off error there, we're gonna avoid that by using the trigonometric ratio, but either one is OK. And we'll get you really, really close to that correct answer. All right. So now we're gonna relate the opposite side again, my to the hypotenuse M and that's done through sine. OK. So we have sine of our angle theta which is gonna be equal to the opposite side divided by the hypo. So 48 degrees is going to be equal to 8.5 m divided by the hypo which is N or the magnitude of the vector, we are looking rearranging just like last time M is going to be equal to 8.5 m divided by sign of 48 degrees. And when we work this out on our calculator, we are going to get that M is about 11.438 m. All right. So we go back to our answer choices. We had already narrowed it down to B or D. Now we have that the magnitude of M is 11.4 m approximately. And so the correct answer here is going to be option D, the X component is 7.65 m and the magnitude of M is 11.4 m. Thanks everyone for watching. I hope this video helped see you in the next one.

Vector A has y-component Ay = +9.60 m. A makes an angle of 32.0° counterclockwise from the +y-axis. (a) What is the x-component of A? (b) What is the magnitude of A?

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Key Concepts

Vector Components

Trigonometric Functions

Magnitude of a Vector