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(III) You are given a vector in the $x y$ plane that has a magnitude of 90.0 units and a $y$ component of $-55.0$ units.(a) What are the two possibilities for its $x$ component?(b) Assuming the $x$ component is known to be positive, specify the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points cntirely in the $-x$ dircction.

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a.$\pm 71.2$ unitsb.$-151.2 \hat{\mathrm{i}}+55.0 \hat{\mathrm{j}}$

Physics 101 Mechanics

Chapter 3

Kinematics in Two or Three Dimensions; Vectors

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our question says you are given a vector in the X Y plane that has a magnitude of 90 units and a white component of minus 55 units. Dennis says A what are the two possibilities for its ex component and be assuming mass component is known to be positive, specified the vector. If you add it to the original one, I would give a result of vector that is 80 units along and point entirely in the minus X direction. Okay, so I wrote down here what we're given at the beginning of the problem were given that the magnitude of Vector V is 90 units and them the white component of that is minus 55 units. So then for part A, it wants us to determine the potential value for Visa. Why? Well, let's go ahead and use the fact that if we draw on Vector V, we know that it has an X component in a white component. So this would be visa backs. This would be visa. Why? It forms a right triangle so we can use Pythagorean theorem. Tio Tio find the ex component. So v squared, According to the staggering dirham, is equal to these of X squared. Plus, he's a y squared. Thus piece of X is equal to the square root of these squared minus BC y squared. Okay, well, let's go ahead and plug in our values for BND supply. So we know this now is gonna be the square root of 90 squared Linus. That's right. Our units, this is 90. You square minus minus 25 you squared. And if you pulling these into a calculator and you carry this out, you get plus or minus 71.2 We get plus or minus because we took the square root and you don't know before the square root if there was a plus sign or a minus sign since both would give you the same answer afterwards Yes. Now we determined the two possible values asked to determine for part a part b then going to go ahead and get a new page here. Part B asks us to then determine what vector would need to be required for us to add it to our original vector. So we're going to call this new vector Victor, eh? And give us a solution that has a value of that era solution. That's a vector with the value of minus 80 or 80 units in the minus X direction. So 80 units in the minus X direction is minus 80 units in the plus X direction. We can represent the ex direction as it's this common notation of I had that's always represent components of your your access. So we want something that gives us this value well. There's also a non arbitrary zero for the Y direction or its other communication. J hot direction. Okay, so we need to determine a CE of X and ace of why to give us this These values, right? So we know a visa, vex and visa by our we have 71.2 for Visa Vex, which is I had direction. Maya's 55 for Visa. Why? Or J had direction. And then we want to see figure out a sub X that we need to be added to this which would be in the high hat direction and the ace of why which would need to be added to this could be in the J had direction to give us our previous answer of minus 80 me. I had direction. American, right. The zero. Because we know that that's there. Okay, so eh, So why, then? I was going to be the ex component of the right hand side of the equation. Minus V selects or 80 minus 80 minus visa back switches. Send me 1.2. Miss is equal to minus 152. Okay, well, we're gonna do the same thing for a sea of why, Ace of why the right hand side of the equation is 00 minus the left hand side of the equation, which is minus 55 minus and minus is plus zero plus 55 which is equal to 55. Let's go ahead and include our unit's on these Israel and values of unit. Right? Okay. No, we are then able to determine Excuse me. Perfect, eh? Vector A is equal to the ex component, which is minus 152. Units in the eye had direction plus B 55 units in the J have direction. Fine box. That is your solution. Purple

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