Vector A is 2.80 cm long and is 60.0° above the x-axis in the fir...

Question

Vector A is 2.80 cm long and is 60.0° above the x-axis in the first quadrant. Vector B is 1.90 cm long and is 60.0° below the x-axis in the fourth quadrant (Fig. E1.35). Use components to find the magnitude and direction of (b) A - B In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch. (Image)

Accepted Answer

everyone welcome back in this video. We have two vectors. So we have vector D. In the first quadrant here and we have vector E. In the fourth quadrant. Okay so vector D. Is 4.2 centimeters long and it's 40 degrees above the X. Axis, vector E. Is in the fourth quadrant 2.4 centimeters long and 40 degrees below the X. Axis. And we're asked to find the magnitude and direction of D minus E. And we're told to use components. Okay. Alright so let's think about what D. Minus E. Is gonna look like. Um So when we're talking about D minus E we can think of this as D. Plus negative E. Okay well what does negative E look like negative E. While we flip the vector E. Okay. And then when we're adding we can add tip to tail. So if we were to add negative E. To D. Would be over here. Like this. Okay. Making our result in vector D minus E. This red one here. Okay so that's just a little idea of what the vector looks like. Now let's get started with our components. So we have the vector D. And we'll just draw it out again so we can see it clearly. Okay we know that this is 40° With the hypotenuse or length of 4.2 cm and we're gonna label this dy okay so the D vector, the Y. Component of the vector and then D. X. The X. Component of the vector. Alright well we can write coast of 40 degrees. Okay. Coast of 40 is going to be the adjacent side divided by the hypotenuse. So we're gonna have D. X. The X. Component of D divided by 4.2. Okay. And that's gonna give us D. X. We can multiply coast 40 degrees times 4.2. That's going to be 3. centimeters. And similarly in the Y direction. This time we're gonna have sign that's going to give us the opposite side which is dy divided by 4.2. Okay. And then multiplying, we get Dy is equal to sine 40 times 4.2 which is gonna give us a value. D. Y equals 2.7 centimeters. Okay, so that's this quantity. So we found here, Dy we found D. X. Alright. Doing the same thing for E. E points down like this. Okay, it has a magnitude of 2.4 centimeters 40 degrees. And similarly will label here. So we have the Y. Component of the E, vector, E. Y. And the X. Component of the vector. X. All right. Now, when we're looking at angle we have the angle here but we want to think about the angle from the positive X. Axis positive X. Axis going counterclockwise. So that's gonna be 360° -40°. Okay, so our angle is going to be 320°. So when we're talking angle we have cose 320°. And this is important in terms of sine. Okay, if we use 40 or if we use 320 we're gonna get the same magnitude but it's important to get that sign right. So we have 320° is going to be EX over 2.4. Okay, let me rewrite this with the The way we've written it can be a little confusing. So let's write E. X is going to be coasts of 320°. Okay. Times 2.4. The hypotenuse of the magnitude. Okay. That's gonna give us a value of 1.8385 And similarly in the Y direction. This time we have signed of 320° times again, 2.4. The hypotenuse of that magnitude we're getting in this case negative 1.45427. Okay. And centimeters are units in both cases and that makes sense. The negative ey makes sense because he is in the negative Y direction. Ok. It's pointing downwards. Alright, so now we've figured out our four components. Let's put them into our table. Okay, so we have X direction. We have the Y direction. We have our vector D. We have our victor E. Okay. And we want to find the resulting d minus E. Alright, so we have 3.2174. We have 1.8 D. Y. We have 2.7 and E. Y. We have minus 1.54 to 7. Okay. And now when we do our components, we add the components, the X components and the Y components. And actually in this case we have a subtraction. So we're going to subtract. So subtracting within this column, the X column we have 3.2174 minus 1.8385. That's gonna give us 1.3789 K. Similarly, in the Y direction, we're going to subtract within the Y column 2. minus negative 1.54 to 7. Well, that's going to give us four .24 - seven. Alright, so now we know the X and Y components of d minus E. Okay, so if we want to draw a d minus E. Well, we know we have a positive X component and a positive word component. The magnitude of the X component is 1.3789. The magnitude of the Y component. 4.24 to 7. Okay. And the hypotenuse here connecting the two. Well, that's gonna be the minus E. That length will be the magnitude. We have our angle theta. Okay, now we're looking for magnitude and direction. Okay, so let's scroll down a bit. So we have some more room. Okay, so the magnitude of the subtraction D minus E is going to be again it's the hypotenuse we can use pythagorean theorem so d minus east squared going to be 1.3789 all squared plus 4.24 to squared. That gives us the magnitude of d minus E squared is equal to 19.9. Taking the square root. Okay. And we're going to take the positive solution here because we're looking at magnitude, we just care about the magnitude. So we're taking the positive route 4. cm. That's alright. So we found the magnitude of our subtraction And now we need to find the direction. Okay, so we know that 10 of that angle theta that's going to relate the opposite and the adjacent side. So we're gonna have 4.24-7, divided by 1.3789. Okay. And we can take the arc tan on both sides to get data equals to an inverse 4.24- 3789 in the denominator, that's gonna give us a data value of 72°. Alright, so there we have it we have, the magnitude of the difference is 4.46 cm with the direction of 72°. It's going to correspond with answer. Be Thanks everyone for watching

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