For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (c) the vector difference A - B

Question

Vector diagram showing vectors D, C, and angles for adding vectors by components.

Vector diagram E1.24 with vectors A, B, C, D, and angles for vector addition.

Accepted Answer

Welcome back everybody. We are given two vectors and we need to find c minus vector D. I'm actually gonna rewrite this and we're going to rewrite this as C plus the reverse of d. But what is the reverse of D? Well, the reverse of D is going to be a vector that has the same magnitude as D but it goes in the completely opposite direction. So this is what the reverse of D is going to look like. So let's go ahead and then do this vector edition right here. So starting from C, we are going to then add on the reverse of D. So it's probably going to go about two right here. This is a reminder, this is going to be the reverse of D. Now we are going to use the tip to tail method. So starting from the origin which is where we started at the beginning of C. We're gonna draw a vector to the end of D. And this is going to be our C plus minus d vector. I'm actually going to call this vector R. Now we are asked to find the magnitude and direction of our so we're gonna need some important formulas for this. The magnitude of a given vector A is going to be given by the square root of its x component squared plus its y component squared. And the direction represented by the angle theta is going to be given by the arc tangent of its y component over its x component. But what are the y and x components of our while the x component of our is just going to be the sum of the X components of our two vectors. The exponents of C and the X component of the reverse of d. Same thing with our Y component of our is going to be C Y plus the Y component of the reverse of D. Let's go ahead and find our our X and our Y. So our X is equal to C. X. Okay, so we are looking for this guy right here so we are going to have the magnitude of C times the cosine of our angle, No sign of 50. And this is going to be negative since we are going in the negative x direction plus the X component of our reverse of D. So we are actually looking for this guy right here. Same thing here. We're gonna have the magnitude of D which it has the same magnitude and then we're actually gonna need to work with an angle here. So something important to establish here, we have two sets of parallel lines meaning that this angle 60 is going to be equal to this angle down here meaning this angle is also degrees. Great. Now with that knowledge will be able to take the sine of that angle and that gives us our X component and it's positive because it's going to positive X direction, plugging this into our calculator. We get that the X component of our r is 4. m. So now let's go ahead and find our Y component here. Same process. We have C. Y. So we are looking for this guy right here. We have the magnitude of C. Times the sine of our angle this time. And it's going to be negative since we are going in the negative Y direction plus our Y component of our reverse of D. We're gonna have the magnitude of D times the cosine of our angle. And this is going to be negative since it is going in the negative Y direction, plugging this into our calculator, we get negative 16.19 m. Now that we have our X and our Y. We are ready to find the magnitude and direction of our So the magnitude of our we're going to use this first formula up here we have is the square root of 4. squared plus R. Y. Component of negative 16.19 squared extend the radical there. And when you plug this in we get that are magnitude is 16.78 m. Alright, so now let's go ahead and find the direction of our our. Now this is not to scale and that'll that'll be relevant in just a second. So our theta is the arc tangent of our Y component 4.41 over our exponents of negative 16.19. When you put this into our calculator we get negative 15.24. But is this the angle that we're looking for this negative 15.24 is actually this angle right here. But the angle that we want is the counterclockwise angle from the X axis. So we are are real data is going to be 100 and -15.24 and this is going to give us 164.76°. So now we have found our magnitude of our summation and the direction of our summation, which is our resulting in our answer choice of B. Thank you guys so much for watching. Hope this video helped. We will see you all in the next one.

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (c) the vector difference A - B

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