Let A = 4i - 2j, B = -3i + 5j, and E = 2A + 3B. (c) What are the magnitude and director of vector E?

Video transcript

Hey, everyone. So this problem is working with vectors. We're asked to work out the vector sum O or O equals three P plus Q using P equals three I minus J and Q equals negative four, I plus six J express express vector O using magnitude and direction form. So the way that we're going to do this is find our components of O in the I and J directions and then solve for the magnitude and the direction using those components. So our multiple choice answers here are a 5. with an angle of 59 degrees above the positive X axis B 5.10 with an angle of 79 degrees below the positive X axis C 5.83 with an angle of degrees above the positive X axis or D 5.10 with an angle of 79 degrees above the positive X axis. OK. So first, we are going to write out our vector sum O in component form. So O equals three P plus Q Will break out three p equals three times P To those components, three I -J. So we'll multiply that three through and three P equals nine I minus three J. OK. So from here, we can add Q, which was given as negative four I plus six J. So for our vector sum O, we will add the I components. So that would be nine I Plus -4 I and then we'll add the J components and that would be negative three J plus six J. Simplify that we get five I plus three J. So now we need to find the magnitude in the direction we can recall that the magnitude is the square root of the sum of squares. So O equals the square root of five squared plus three squared And that equals 5.83. And then for our angle given that we have our X and Y direction, that will be our opposite and adjacent sides, which is our hint to use tangent. So theta is going to equal the inverse tangent of our opposite side. That's three over our adjacent side, that's five. And that comes out to a positive degrees. And so when we look at our answer choices that aligns with answer choice C magnitude of 5.83 and an angle of degrees above the positive X axis. So that's all we have for this one, we'll see you in the next video.