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(a) $P_{1}(3,5), P_{2}(2,8)$ (b) $P_{1}(7,-2), P_{2}(0,0)$ (c) $P_{1}(5,-2,1), P_{2}(2,4,2)$ (a) $P_{1}(-6,-2), P_{2}(-4,-1)$ (b) $P_{1}(0,0,0), P_{2}(-1,6,1)$ (c) $P_{1}(4,1,-3), \quad P_{2}(9,1,-3)$

In this problem, we were asked to find the center of mass of the two dimensional system with masses 2 34 to 1 in six in the positions. Negative too negative. Three 55755100 and negative 30 In order to do this, we'll need to calculate both this center of mass in the extraction, which is given by the moment of why, over the total mass and then the center of mass in the UAE direction, which is given by the moment of action over to the trouble of mass First calculating X bar, the moment of wise given by each of the masses. Times there X coordinates allotted together so we can find that by three times negative too, plus four times five plus to turn someone plus 1 10 1 zero plus six times negative three all over the total mass is going to be three plus for someone close to his nine plus one is 10. Post six gives you 16. So doing some arithmetic we get three times a year to is negative. Six plus 20 is 14 plus 14 is 28 poses zero post negative 18 is going to give you 10 over 16 or 1.6 now to find the center mass in the wind direction, we're going to find Calculate the moment of X, which is given by the masses times there y components. So we have three times negative. One negative, three plus four times fine, but two times one plus one time zero was six times zero all over the total mess, which is still 16 now doing our athletic three times three makes nines with negative nine plus 20 which is this 11 plus two gives his 13 and both of these terms to goto zeros. We get 13 over 16. So are finally answer is export y bar it wa ce 10 over 16 comma 13 over 16.

We have this flow chart and we have to find the output of this flow chart. So let's start executing this flow chart before we execute each operations. When we want let's try to understand what exactly this flow chart does. It starts with the start command, then we have some operation box. We're in the values of A. And the ceo initialized and then some genius to diseases squired and then added with A. And is in fact doubled over here. And then it checks whether the value of a is still less than eight and when it does it will go ahead and or do these operations once again. So basically we have a loop over here It gets repeated as long as the value of a. is less than eight. once the value of is not less than eight. That is If a is either eight or anything more than eight like 9 then the local stop and it will start printing the value of C. So let's start executing this one x 1. So initially the value of the equal to one & C Equal to two. And when it comes over here the sea is getting updated by squiring the value of C. And then added with eight. So when we square the value of C we get to square is four. And when you add with a that is one we get sequel to fight. So therefore the current value is equal to five. And the next statement we have A is doubled. So equal to a start to, this is basically is getting double Which means this is the initial value of eight and double it will get to. So therefore these are the two values which we have done now. That is equal to five and equal to two. Well then how this decision box which checks whether it is still less than eight, Which is in fact true, that is two is less than eight. Yes. So that means the control will be shifted back to this point which means we are into the next to loop. So in the next two loop, let's write down the current values of AIDS and sees. So is already updated too two and she is now updated to fight. So these are the values that we are going to it place when we place this look. So then once again she's getting updated by squiring the sea and adding with a which means the sea will be square. That is five x squared. So therefore if you do that it will be 25 Plus getting added with a. That is too. So this gives value. See the new value of sequel to 27 And once again is getting doubled. So this is the value of a. That is too when a double two will get a call to four and once again we check whether still If a is to less than eight so is less than eight. Yes this is is therefore the control will be shifted back to the this point which means we are into the next loop. So let's write on the existing values of A. Or the updated values of A. And C. You know that this is the an updated value of we put here And c. is 27 so we put the C over kid And then once again the sea is getting squired and added with a. Which means c equal to 27 Times 27-plus 4. So when it's choir 27 This is equal to 729 plus food Which is equal to 7:33. Therefore the updated value of c equal to 733 and still and then the next statement we held this getting doubled as usual which means this four is getting doubled so therefore equal to eight now. And we have this decision is eight less than eight. No it is not. So therefore the loop will stop over here. It will come out of the loop and it will execute the next statement which means it prints the value of sleep. You know that the greater values equal to 77 33. So therefore at this point the computer will print the value 733 which means this option is correct

In this problem. We asked our ass to find the center of mass of this two dimensional system with the masses. 2 8 to 1 and four in the positions. Negative three negative. 100 And negative one, too. In order to find the center of mass, we first need to calculate the center mass in the ex direction, which we will do by calculating the moment in the UAE direction divided by the total mass. Once you've done that, we will calculate the, um, center of mass in the wind direction. Just calculate the moment in the ex direction of over the center of mess. Okay, So calculating Expert, um, the moment of wise calculating by multiplying the masses by the X coordinates. So we have eight times negative. Three plus one time, zero plus four times negative. One all over. The total mass, which is given by eight plus one is nine plus four gives us 13 doing some arithmetic. We can find out that eight times negative three is negative. 24 0 minus forgives us negative. 28 over 13. Now, to find the center of mass in the UAE direction, we're going to take the moment of X, which is the masses times there. Why Coordinates? So we're going to have eight times negative. One plus one time, zero plus four times two all over the total mass, which is still 13 here. Only two are particularly. Get negative 80 plus h, which gives us zero over 13 for our center of mass. Explore why BAR is given by the coordinate point. Negative 28 over 13 from a zero.

To find these two matrices together. Here. First thing we need to do is check the dimensions will work. Okay? First matrix we've got is two rows three columns. That's a two by three. The second matrix is a three row one column matrix. These inner two dimensions have to match which they do. That means is a resulting matrix will be dimension the outer dimensions. Okay, So we're gonna have to buy one. Matrix is our answer just two spots here. Okay, we'll fill in the first spot. This is gonna be row one. Column one. Okay, so everything in row one thinks everything in column one negative one times six plus zero times negative four plus seven. Thanks one. Okay, this should give us negative six plus zero plus seven. That should give us a one in that first little spot right there. Okay, The second guy this is gonna be row two column one. Okay, so we're gonna take everything in row two times that same column right there. So three times six less negative five that was negative for plus two times one spent 18 plus 20 plus two should give us 40. Okay? So one in the first row, 40 in the second round, and that is the result of this modification.