5

| 1 3 H IHK L 1l M 1 H 1 H V 4 W { 3 8 0 8 WH | X 1 1 1 I 3 1 3 } Ji Ti 1L 1 8 U W 1 1 1 3 IY ! 1 1 L 1 3 1 1 1 U | L 1 8 L 1 1 Vi 2 3 6...

Question

| 1 3 H IHK L 1l M 1 H 1 H V 4 W { 3 8 0 8 WH | X 1 1 1 I 3 1 3 } Ji Ti 1L 1 8 U W 1 1 1 3 IY ! 1 1 L 1 3 1 1 1 U | L 1 8 L 1 1 Vi 2 3 6

| 1 3 H IHK L 1l M 1 H 1 H V 4 W { 3 8 0 8 WH | X 1 1 1 I 3 1 3 } Ji Ti 1L 1 8 U W 1 1 1 3 IY ! 1 1 L 1 3 1 1 1 U | L 1 8 L 1 1 Vi 2 3 6



Answers

(a) $\mathrm{F}_{1}(\mathrm{a}) \mathrm{F}_{1}^{\prime}(\mathrm{a})+\mathrm{G}_{1}(\mathrm{a}) \mathrm{G}_{1}^{\prime}(\mathrm{a})$ (b) $\mathrm{H}_{1}(\mathrm{a}) \mathrm{H}_{2}(\mathrm{a}) \mathrm{H}_{3}(\mathrm{a})$ (c) $F_{r}(a) G_{r}(a) H_{r}(a)$ (d) 0

They're. So for this exercise we have this vector B. And the subspace dovey generated by the one, V two and V three that are these vectors that are defined here. So basically we need to calculate the Earth a little projection of you on this space to view. And just remember remember this projection is calculated as the inner proud of the vector V. Each of the generators of this subspace dog. In this case the generators RV one, The two and 3. So we need to calculate the we need to calculate the inner part of me with each of the generator divided the score of the norm of the generators times degenerates. So these for the three vectors B two square plus the interpreter of B would be three. B three. Did the square of the norm of B. Three. Okay, so just to remind you a little bit of the geometric intuition of this, is that the view is generated by these three vectors. So what we're doing is projecting we on each of the generators and then some that together. So we want We t. v. one and V three acts as a basis. Actually in this case they are linearly independent so they form a basis for this. Yeah, subspace of you. So we're writing the in terms of this basis. So we're projecting projecting on this sub space. So let's calculate the correspondent values that we need. So in this case we would be one. The product of B would be to dinner product of the would be three. So this is equal two, one half, There is a constitute and this inner product is equal to zero and then the norms. So because this is the cost to zero means that we don't need this term anymore is going to be equal to zero. So we just need to calculate the score of the norms for B. two and B one. So for me, one square of the norm, remember that there is equal to the inner product of the vector with itself. And in this case this result in one and the inner approach of B two square is equal 2, 1 as well. So these are actually military vectors. And then we just need to put all together on the four. So behalf that the projection of the vector B on the subspace, our view, it's equals to 1/4 times 11 one plus the vector V two. That is equal to one, 1 -1 -1. After some. In these two vectors obtain the action solution that is one half times the vector, three, three minus one minus one. That corresponds to their thermal projection of beyond this subspace of you.

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.

Option. We are given an equation which states H two S. Which is engaged. His home next to products, which is a chess gas plus edge gas. Now we need to find the intel P of formation. So until P of formation, delta edge can be written as the intel P. Of and help of product minus and help me off and help you of reactant of reluctance. Here we have two products Hs and edge and a reactant H two S. So we can right here that delta edge of Hs plus delta edge of edge minus delta edge of H. Two S. Institution. We are given that and help or formation is equal to X one. And and tell P of H is we need to find let that be A. So it can be written as excellent equal to a plus delta edge. And help you of edges given as X. Tree and and help U. Of H two S. Is given as X two. So from here I can write is going to be X one plus X two minus X. Tree. So option is going to be the right option

So in this problem we're given this five x 5 matrix And were asked to use a matrix calculator, luckily in order to determine or find the determinant. All right, because you can do this by hand. I might be a little difficult. You can probably pick Grow four here and you know avoid two of the calculations. But then you're gonna still be left with four x 4 Matrix sees and working those ways where's goes down. So this can be a lot quicker. So we go over here to the new matrix. In our matrix calculator. We went to Desmond's dot com math tools matrix calculator to get this. And our matrix is a five by five. I gotta get this thing built up here, the framework of it five by five. and so the first entry is a three and then a minus two A four three. You know, one, Then a 1-0 -1 zero 2102 on Euro. And then 5 -15. My ass. One 032 zero three, two Than 4 7 -8. 4 seven. Mine it's eight and zero zero. And then we have 123 one two three 02 zero two chris Sanders. So that's all in there. Now I need the determinant of this Five by five. So I go d et determinant of A And there it is 410. That was a lot quicker and a lot easier. And trying to do this all out my hand, wasn't it


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