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What is the answer please??...

Question

What is the answer please??

what is the answer please??



Answers

what is the answer?

In this problem, equation of two planes are given as two x minus three way plus kids did is equals two for and the second plane is ky minus did plus three equals to zero. So we can send this plus three times in the right hand side. So this will be minus three. So these are the two planes let us say this is plain one and this one is plain to. And in the problem it is asked to that, what are the values of K for which the two planes are parallel and two planes are perpendicular. So first we will solve for them perpendicular planes. So we know that when two planes are perpendicular in that case, uh their normal vector will also be perpendicular. When when equation of plane is written in car, in Cartesian coordinate system as eggs plus B, Y plus seeded is equals two D. Then coefficient of A B and C represent us. The coefficient of X. Y, injured represents normal vector of that plane. So normal vector for this plane can be written as E cap plus B J cap plus C. K. Cap. So are using this concept, we can write normal vector for plain one. So normal vector for plain one. Let us say anyone vector will be equals two to icap minus three. Jacob plus ki kick up here is the constant. All right. Ah Similarly, we can write the normal vector for second plane. It will because too, since there is no x coefficient, therefore I kept coefficient will be zero. Only there is why injured involved. Therefore, there will be a key Jacob minus one K cab. Now, since these two planes are perpendicular, then uh these two vectors will also be perpendicular and when to victor's are perpendicular there dot products will give will give zero therefore anyone vector dot with And to victor must be zero. Now, what is anyone victor? It is to icap minus three Jacob let's key key cap dot with kg cab minus one. Key cap, this must be zero. Now, how do we solve a? How do we solve a dot product? So we multiply I kept coefficient with icap coefficient here, I kept coefficient is too and in the right in the second victor there is no I kept coefficient. Therefore this is going to give zero plus. Now we will multiply Jacob coefficient with the Jacob coefficient. In first vector Jacob coefficient is minus three and in second vector it is key. Now we will solve the third uh coefficient. So K cup coefficient is key. And as in second case, Cape cod coefficient is minus one. So this is going to be zero, so minus three K minus K must be zero. Therefore minus four K must be zero. So, key value of K. Four ways that two planes will be perpendicular is zero. So we have got the value of K. For which the two planes will be perpendicular. Now we will solve for the parallel planes. So the second cases parallel planes. No uh we will use the same normal vectors again. So I'm copying these normal vectors since normal victor of the plane will remain same as the equation of plane is same, but not the condition had changed. Now these two planes are parallel, so when two planes are parallel then their normal vectors are also parallel. And for two normal vectors we know that they're cross products must be zero. Therefore anyone victor cross into victor must be zero. So how do we solve our cross product? So we write it in determinant form. Now the first three columns is Icap, Jacob. And kick up. we will write first normal vector coefficients in the corresponding column. So I kept coefficient of anyone victories to Jacob coefficient of and two of anyone victories minus three. And K cab coefficient is okay. No. What is the I kept coefficient of anti vector? It is zero. Jacob coefficient is K and K cab coefficient is minus one. And value of this determinant must be zero. All right, so uh how we will solve this if we expand this determinant. So first we will write I kept and then minus three, multiplied by minus one and then came negative with K multiplied by K. So minus three multiplication with minus one will give three minus K esquire. Now we will write minus Jacob as the sign convention is plus minus and plus so minus Jacob and then four minus Jacob. We will multiply two with minus one, so it will give minus two minus with zero into K. Is going to give zero not get so efficiently K cab coefficient will be our two multiplied by key and then we will subtract zero, multiplied by minus three. So it will be to k minus zero. So this must be zero. So what is the icap coefficient here? So I kept coefficient is three minus K square. I kept plus to Jacob and plus two K cab. This must be zero plus to take a cab. Mhm Plus two K K camps must be zero. Now we see here that a coefficient of Jacob is a constant. So whatever we put the value of it is not going to affect the Jacob coefficient. So it will always remain in left hand side, but in right hand side it is a null vector. So we can write zero as zero icap plus zero Jacob plus zero K cab by putting their different values of K. We can we can get the value of Icap coefficient zero. We can get the coefficient of Jacob also zero. Since that is also dependent on the value of K. But we need to check that is both. Both. The coefficient must be zero at the same value of K. But whatever be the value of K coefficient of Jacob will never be zero since its value is too. Therefore there will be no solution in this case. Therefore no such value. No such key exists for which the planes I'll parallel. So it's correct answer will be option. See that is key equals to zero. That is for the perpendicular case. Okay, see cake was 20 and no such value. So this is the answer for this problem.

We have a question. Ah number seven in which we have to find the value of care that will make the planes first plane is two X minus kyi plus trees said minus one equal to zero and two K X plus three Y minus two. Said equal to four perpendicular. Okay, so if you compare this with planes even X plus B. One, Y plus even ZD plus even equal to zero will be getting uh even B 17 equal to two minus K. Three. These are the normal vectors. And here similarly if you compare with eight weeks, be too wise. It was that? So a two B two C two will be equal to to care comma three minus two. So these two will get will be perpendicular. These directors perpendicular If the dot product of their normal vectors are equal to zero. So if this is normal vector, anyone this is normal victor and two. So anyone dot and two should be equal to zero. Which means this is the X component. Why is that X. Y. Is that? So we will be having two and two. Two K plus minus cane +23 plus three to minus two should be equal to zero. Okay? And that is four K minus streak minus six should be equal to zero, so case should be equal to six. So for these two plans to declare the value of case should be equal to six. So she is the correct answer. Thank you.

We have question number three in which we have to find the value of K. That replaced the point which is given on the line. Okay, so this is not a director equation of the line minus three comma two comma nine plus as three comma to common minus four and s belongs to our that is real number. And ah we have to place this point K comma two K comma K minus two. So for this point B to lie on this line on this line we should be writing this as K comma two. K comma k minus two equal to minus three, comma two comma nine plus S three comma two comma minus four which means. Okay comma two K comma K minus two should be equal to minus three plus three years minus three plus three years. Okay comma this is too bless. This is to us comma this is nine and minus for us. Oh now we have to equate each and every turn to get the value of K. And as so first this K will be equal to minus three. Last three years. Second term to K will be equal to to place to us. And the third term k minus two will be equal to nine minus forests. Okay so from here we have we will be using any two of these questions. There's three questions. I need two of these three questions to get the value of K and S. And then we'll be plugging in third equation by to check if the values are correct or not. So let us start here since we need to find the value of. Okay so value as we must find out. So this value of K. So let us plug in the value of cane equation number two, two minus three plus three years equal to two place to us. So this will become if you take two has come and it will become one process. So two and two will get cancelled out. So this is minus three plus three is equal to two place to us. So as will be equal to yeah five. Okay. No. Yeah. Okay. Okay. There is a mistake that I have already cancelled out this too and again using it. Okay one less. S and now two years will be equal to four. So from here as will be equal to four by two. That is to so we have a sequel to two. So from equation number one we can get the value of K. So they will be equal to minus three plus three into two, that is minus three. Bless six so equal to three. So we have value of K. Three and value first to now let us plug in in this equation. Ah uh we'll be playing the value of K. N. S. And let us see if they satisfy this equation or not three minus two. Because K is equal to three from here and as his duel, so nine minus four into 21 equal to nine minus eight, so one equal to one which is true. So the value of K we satisfy this equation is three. An option number A should be the answer. Thank you.

So here in this problem, we are given that the magnitude of the vector A. Is four and we have to calculate that. What is the dot product of the victor? A with the victory. So let us take this be the victory. Let this directory and its magnitude of the vector A. We have given us for now. If you use the rule of a dot, right, we know that A dot be is equal to mode A. That is a magnitude of eight times the magnitude of pay times. Cosine Theta where theta is the angle between A and B. So if you use the same formula here, a dot A will be called to the detriment of A times the detriment of times the cosine theta. Now this is a victory. There is the angle between the vectors A and the vector is zero. This would be zero. So this would be called to a square and a square means foursquare, which will be equal to 16. Therefore a not a S. 16. Therefore the option B is correct. Option B is correct. So I hope you have understood the problem. Thank you.


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