 चोरी का दोकिर लीरा ती ता ऐ nost alo a2 अधन �見 भित Eyes छग टरी kini aur bandare ka kini और करद ठह को दो हूणर हacle koi तpeak खॉरीब स्थमनी। तpeak खॉरी घीं थ supreme तpeak खॉरी पर करना उस कस जीता विम में. स्थमनी onка दii gurii ूरी । तpeak तpeak 2 i minus 3 j plus 5 k plus 7 equal to 0 right so i can say that the normal for this also will be 2 i minus 3 j plus 5 k or at least a scalar multiple of that correct so i can say that my desired required equation would be r dot 2 i minus 3 j plus 5 k okay equal to some constant which is not 7 in this case so some some value you can write it let's say i call it as you know value l okay now to find this value l only you have been provided that the plane passes through 3 4 minus 1 correct so we can put that over here so instead of r put 3 i plus 4 j minus k dot 2 i minus 3 j plus 5 k equal to 1 okay so just open the dot product over here it'll get 6 minus 12 minus 5 is equal to l is equal to l so i think this gives you minus 11 as your l am i right yes or no back over here so my final equation was down to r dot 2 i minus 3 j plus 5 k equal to minus 11 or plus 11 equal to 0 so this becomes your desired answer so but can we take the normal to be same and not scale up multiple does it matter though if you take it yes i didn't take that r dot only yeah it doesn't matter see why it doesn't matter let me show you how so let us say i had given you r dot let's say 2 lambda i minus 3 lambda j 5 lambda k okay equal to some constant correct yes or no so when you put the point here you end up getting minus 11 lambda is equal to l right correct so when you put this back over here you'll get minus 11 lambda drop lambda from everywhere now you are back to the same equation which i got is it because our direction ratios are infinitely many so yes yes and see normal can be from here normal can be from here normal can be from here normal can be from here any point parallel to each other right so normal doesn't mean it has to pass through the center of a plane there's nothing like center of a plane or something normal is a plane is an infinitely extending geometrical figure so any perpendicular to the normal whether both the normals are parallel to each other it doesn't matter to us yes make sense make sense okay fine so next we are going to talk about one of the very important concepts which is called the concept of line of intersection line line of intersection of two planes in short form I'll refer to it as L O I okay so when two planes when two planes intersect okay when two planes intersect they intersect on a line so this red is line is called the line of intersection this then line is called the line of intersection okay so many a times when we have to mention the equation of a line they will just mention you the equation of the two planes and they say this is the line for example they'll say there's a line given by the equation x minus y plus 2 z equal to 5 and 3x plus y plus z is equal to 6 okay let's say they'll give you like this okay and many people they get surprised they say hey they have given the equation of two planes and they're claiming that it's a line equation yes it is a line equation all they mean to say is that the line is created by the intersection of these two planes are you getting my point so mentioning such equations is called the asymmetric form of the equation of a line this is called the asymmetric form of the equation of a line so whatever we had learned was actually the symmetric form that is x minus x1 by a equal to y minus y1 by b equal to z minus z1 by c but this is called the asymmetric form so don't be surprised in future if you see there's a line equation which is mentioned as the equation of two planes is just another way of writing the equation of a line which is basically passing through the intersection of those two mentioned planes have I made myself clear okay now what type of questions will you get the two types of question which is based on it one is converting the asymmetrical form to symmetrical form that's one type of question and and second type of question is there would be some information about a plane passing through the line of intersection thereby giving it a family of plane concept so we'll talk about the second concept little later on let me first take a question where I have to convert asymmetric form okay let's take this as question itself so this is an asymmetric form of a line tell me the symmetric form for it now since you have not done any question of this type before let me start with this question as a illustration problem so when you want to find the equation of a line what are the two things that you need you need the direction any line you need a point and you need a direction right right without these two things will not be able to get the equation of a line correct now if you want to find a point what do we do is we put any one of the three variables as zero in both the equations okay so let's say I put zero a z as zero in both the equations so put z as zero in both the equations so you'll end up getting x minus y is five and three x plus y is equal to six okay so from here you can find out the value of value of x to be eleven by four okay put it in any one of them let's say I put in the first one so why eleven by four minus y is five so y is equal to minus nine by four okay so we have got a point on this line which is eleven by four minus nine by four and zero now people ask me sir could I could I put y also zero yes you can could I put x also zero yes you can it's your call okay so the point may be different for different different people now here a very important precaution to be exercised is that when you're putting any variable as zero it should not lead to an inconsistent equations per se let me give an example where such thing may arise let's say you had equation x plus y plus 2 z equal to 5 and x plus y minus 3 z equal to 6 right let's say these are the two planes and you decided to put x equal to 0 what will happen it will lead to an inconsistent set of equation isn't it one is saying x plus y is 5 other is saying x plus y is 6 this is not possible you cannot solve this right so refrain from choosing such refrain from choosing such points as zero which will lead to inconsistency in your answer it is just equations way of saying that in on that plane z equal to zero is not possible have I made myself clear yes sir okay second thing is people ask me sir is it necessary to put zero can't I put one or two or something yes you can you have full right to put any value to any point but just that whatever are the other two equations that are coming up should not lead to infinitely many solutions or should not lead to inconsistent solution inconsistent equations okay because we just need one point so here we have got a point so one task is over okay so point is known now how do I get the direction to get the direction it's very obvious that this line of intersection is simultaneously perpendicular to the normal of pi2 and perpendicular to the normal of pi1 isn't it so let's say n1 n2 are the normal vectors I'm sure you would all be able to appreciate that this is also perpendicular and this is also perpendicular so this line will lie on which plane both the planes my dear both the planes pi1 and pi2 it is the line of intersection of both the planes so think as if it is like your you know the corner of your book right and the pages are your planes are you getting it so the line of intersection lies on both the planes it lies on both pi1 and pi2 plane correct and at the same time it is perpendicular to n1 and n2 also do you realize that of course one implies the other if it lies on pi1 then n1 will be perpendicular to the line of intersection right if it lies on pi2 n2 will also be perpendicular to the line of intersection that means n1 n2 both are perpendicular to the line of intersection is that clear this is very important everybody are there are hamsony licketh is that clear yes okay fine so if if I know the normal to both the planes I know n1 n2 vector can I not find out the direction of the line so what is the direction of the line here you will say simple it will be along the direction of n1 cross n2 yes or no any vector which is simultaneously perpendicular to two vectors lies in the direction of their cross product so n1 cross n2 you can find it i j k 1 minus 1 2 3 1 1 okay so this is okay let me expand it so it's becomes minus 3 minus j it becomes minus 5 okay and k will become 4 am I right okay will become a 4 so it's minus 3 i plus 5 j plus 4 k so you can use this minus 3 5 and 4 as your abc of the line so finally concluding your equation of the line will look like this in symmetric form x minus 11 by 4 see 11 by 4 was your point divided by minus 3 is equal to y plus 9 by 4 divided by 5 and z minus 0 by 4 this becomes your equation of a line okay you can want if you want you can write it in a simpler form 4x minus 11 by minus 3 4 y plus 9 by 5 is equal to z minus 0 by 1 okay this is your symmetric form so i'll just repeat the procedure once again what do we do to convert asymmetric form what is an asymmetric form whenever you have been given the equation of a line by mentioning the two planes whose intersection generates that line we call that equation to be a asymmetric form so in order to convert it to symmetric form you need the point and you need the direction of the line to get a point we put one of the variables as 0 1 2 whatever you feel like but it should not make the system of equations inconsistent once you have found out the points now you can find out the direction by taking the cross product of the normal vectors to both the planes and once you know the direction you can always write down the equation of a line does that make sense everyone if that makes sense i will have to give you a question here you go find the angle between the lines now see here there are four planes mentioned in fact there are there are two lines mentioned this is your line number one l1 and this is your line number two l2 right and they're now asking you the angle between these two lines try it out that's correct share anybody else share has already replied guys hope you're not finding the complete equation because i just need the direction ratios of the two lines so don't give me the symmetric form for both the lines that is not required aditya nikhit ardhara hamsini samikta i'm getting the perpendicular let me tell you you are correct aditya okay so all you need to do was get the direction ratios of the l1 line so it's perpendicular to both the planes simultaneously so let's write i j k 1 minus 3 and there's no z over here right so 0 okay and here there's no x so 0 4 minus 1 okay so when you expand it it becomes 3 okay minus j minus 1 and k will give you a 4 so 3 i 3 i plus j and plus 4 k that's the direction ratios you can say 3 1 and 4 so for the direction ratios of the other line l2 again i have to do the same thing i j k components are 1 3 0 and 0 2 minus 1 okay so it's i minus 3 minus j minus 1 and k 2 so it's minus 3 i plus j plus 2 k fine so the direction ratios here are 3 minus 3 1 and 2 so let us find out the angle by using the formula a 1 a 2 b 1 b 2 c 1 c 2 by under root of a 1 square b 1 square c 1 square but let me find the denominator numerator first so a 1 a 2 is minus 9 plus 1 plus 8 that itself becomes a 0 so no need to do any further it implies l 1 and l 2 are perpendicular is that clear this so everyone okay