b. Use geometry and trigonometry to determine the magnitude and direction of G = E＋F.

Video transcript

Hey everyone. So this problem is dealing with vectors. Let's see what it's asking us to describe a vector. We need both its magnitude and its direction, determine the magnitude and direction of the result in vector C when C equals A plus B using the graphical method which is telling us geometry and trigonometry. So our multiple choice answers here R A four with a direction of 28. degrees B 10 at a direction of 13. degrees C 4.29 with a direction of 56.5 degrees or D 2.83, a direction of 45 degrees. So we are given vector A which has a magnitude of three and it is only in the extraction. So it would be three I plus zero J and then we have factor B which is in the um negative X and positive Y direction. So in uh component form, it would be negative one I plus two, Jane, we're also shown uh that B makes an angle of theta with the positive Y axis and then the total vector um or the angle between vector A and vector B is given here by angle um fine. OK. So we're going to drop in our resulted vector, just kind of name that R. And from hour trigonometry, we can continue drawing this parallelogram. And so now using are um again trigonometry rules, we know that R squared is equal to A squared plus B squared minus two A B cosine of alpha where alpha is this angle here between the A vector and a line parallel to the B vector. OK. So these are all magnitudes, these are the magnitudes of the vectors. So we've already established that the magnitude of A is three because there is no Y component, the magnitude of B is going to be the square root of the sum of the squares again a little bit of basic trig there. So that's negative one squared plus two squared squared of that. He's just the square of five. OK. So the last thing that we need to solve this equation for the magnitude of this resultant vector um which I'm calling R here. But I guess the problem calls it C it's the, the same and interchangeable is this angle alpha. That's the last thing that we need. So based on um our, again trig rules, we know that alpha plus five equals 180. So these two angles here, 180 and five equals 90 plus theta. And so another, when we put that together for alpha, we can say that alpha equals 180 degrees minus 90 degrees minus data. And we know that theta, we can solve theta because we have the opposite and the adjacent uh sides of that angle. And so quickly, we know that theta equals the inverse tangent of opposite native one adjacent is two and beta equals 26.6 degrees. OK. So now we can plug that in here 26.6 degrees. And so that leaves us with an alpha of 63. degrees. And now we can plug in everything we need for to find um see the magnitude of this resulting vector. So take the square root of A squared. So that's three squared plus B squared is so B is the square root of five squared minus two multiplied by A three multiplied by B squared or five multiplied by the cosine of alpha or 63.4 degrees. We plug that into our calculator and we get 2.83 as the magnitude of our resultant vector. So when we look at our multiple choice answers, we can see that answer D is the only one that's left. But let's solve this last piece for our um angle of the resultant vector just to make sure that we are on the right track. And so the resultant vector, the angle for that is going to be here, we'll name that angle beta. And so using trigonometry, the law um of sign angles here, we can write that B, the magnitude of that vector B divided by the sign of beta is equal to C divided by the sign of alpha. And so here you can see that's the, uh the side opposite of the angle is. Um so this line here is the same magnitude as B it's parallel. Uh And so that's how that works out. And so this can be rewritten as B multiplied by a sign of alpha divided by C is equal to the sign of beta. And then when we plug in our known values here, so B is squared at five multiplied by sign of 63. divided by 2.83. And we'll actually take the inverse sign of that to get beta that equals 45 degrees. And so that aligns with the answer choice D you have a magnitude of 2.83 at an angle of 45 degrees. So thanks for sticking with me through this one. We'll see you in the next video.