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

Video transcript

Hey everyone. So this problem is dealing with vectors. Let's see what they're asking us using the vectors M equals three, I minus two J and N equals negative two I plus six J give the result of vector O which equals M minus two N using the magnitude and direction form our multiple choice answers. Here are a, a magnitude of 15.7 at an angle of 63.4 degrees below the positive X axis B A magnitude of 12.6 at an angle of 71.6 degrees above the negative X axis C and magnitude of 9.3 at an angle of 63.4 degrees below the positive X axis or D A magnitude of 15.7 at an angle of 26.6° below the positive X axis. OK. So when we are working with vectors, they, they've given us these vectors in component form so that they're telling us in the I and the J, what those magnitude of those components are, they are asking us to report out this result using magnitude and direction form. So we're going to first get the resultant of vector O in component form. And then we will use that to determine the magnitude and direction form. So we know from the problem that vector O equals M minus two N where vector M is given to us as three I minus two J and vector N is given as -2 I plus six J. So we need to get this two N. So we will multiply this two through and get negative four I plus 12 J. So now we have the parts to determine vector O using component form. So we will take the I terms and then the J terms. So the I term for vector O is going to be three I minus the I term from two end. So minus negative four I and then in the J terms, we have negative two J minus 12 J. And so we simplify that and we are left with seven minus minus 14 J. OK. So that is the component form of vector O but they've asked us to uh solve for it in the magnitude and with the magnitude in a direction. So we can recall that the magnitude is the square root of the sum of the squares of the I and J components. That looks like the magnitude of O equals seven squared plus negative 14 squared. We plug that in and we get 15. and our angle when we are given opposite and adjacent from trigonometry that tells us that we're gonna be using tangent. So the tangent of beta equals the opposite over the adjacent and, and then we can write beta equals the inverse of tangent times the opposite, which is negative 14 over the adjacent, which is seven. And that gives us negative 63.4°. So because it is negative, yeah, that tells us that it is below the positive X axis. So our magnitude is 15.7 and our angle is 63.4° below the positive X axis. When we look at our multiple choice answers that aligns with answer choice A. So that's all we have for this one. See you in the next video.