 this question so let me take another one yeah by the way this writing is not that clear so it's x plus 1 by minus 3 is equal to y minus 3 by 2 is equal to z plus 2 by 1 so find the equation of the plane which is parallel to the lines this and this and passing through 0 1 minus 1 now let me tell you this is not different from whatever we have done so far is just an extension of that concept yes definitely so when you say a line a plane is parallel to two lines that means the direction of the normal to the plane is perpendicular to both the lines okay yes anyone any success so see just just try to imagine it just try to imagine the given scenario so you have a plane like this let me call this plane as a pi plane okay and this plane is parallel to the two lines parallel to the two lines means let's say the one line is like this or the line is like this so parallel doesn't mean the line themselves have to be parallel so it can be those both lines could be skew as well okay so imagine as if you have so she has sent me an answer no sure that's not the right answer I'm afraid so these two lines are parallel to this plane okay so they're parallel to this plane so they are above it and they are parallel so think as if there's a fly over which is parallel to a road which itself is made up of two crisscross roads okay are you hold a notebook and just hold a pen on top of it which is parallel to it now you can have two pens I can have multiple pens you know which are parallel to the same plane so can I say in order to get the equation of this plane I would need the direction of the normal to the plane right and this normal will be simultaneously perpendicular let me extend this this normal will be simultaneously perpendicular to the direction of both the lines right yes sir okay yeah that's fine but what is the answer you have given me I think vector this completed you are in the right direction by the way so let's say this direction is B1 direction and this direction is B2 direction so can I say this normal which is the normal to the plane would be in the direction of B1 cross B2 so you can take it as B1 cross B2 itself okay so let's find B1 cross B2 by using our vector product so components of B vector will be 214 components of B2 vector will be minus 321 okay let's expand it so 2 minus 8 sorry 1 minus 8 is minus 7 minus j that would be 2 plus 12 which is going to be 14 and plus k 4 plus 3 which is 7 right so this is normal to the plane now you can retain this normal as it is or you can divide by a proportionality constant for example I can see 7 here is appearing as my proportionality constant so what I will do is I will divide by that 7 and write this normal to be minus i minus 2 j plus k correct it's your call or you can retain the same thing doesn't matter answer will not be affected take my words for it now you know the normal and you have been given a point so let's say I want to write a Cartesian form how will I write it so remember my direction ratios of the normal are provided here which is minus 1 minus 2 and 1 okay all I will use the formula is ax minus x1 by minus y1 cz minus z1 is equal to 0 okay this point is behaving as your x1 y1 and z1 so your answer will be minus 1 x minus 0 minus 2 y minus 1 and 1 times z plus 1 equal to 0 so expanded it becomes minus x minus 2 y plus z plus 3 plus 1 so this becomes your answer okay you can write it as x plus 2 y minus z is equal to 3 also and never be complacent when you get your answer always quickly verify the 0 1 minus 1 satisfied let's check 0 1 will give you 2 minus minus 1 will give you plus 1 so 0 plus 2 plus 1 is equal to 3 yes it is very fine so your answer is correct is that fine everyone again an indirect question but using the fact that you know a point and you know the normal to the plane so once these two information are known to you directly or indirectly you can always write down the equation of a plane by using this formula does it make sense to you yes sir everyone I think Shreya just made a silly mistake yeah I made an addition mistake addition mistake let's try out this question if a plane meets the coordinate axis is at a b and c such that the centroid of the triangle is 1 2 4 find the equation of the plane is the question clear to everyone see the question says there's a plane which meets the coordinate axis is at a b c so let me just draw a coordinate axis is for you okay so there's a plane let's say which meets the x axis at a y axis at b and z axis at c this is a z axis and it meets at c and when you make a triangle out of a b c when it says that the centroid of this triangle a b c lies at 1 comma 2 comma 4 find the equation of the plane which plane this plane okay hope you can understand by my shading which plane I am referring to the plane which contains this triangle any idea how to do this no sure this is not correct you have been given the centroid of the triangle you have not been given the coordinates of a b c oh yeah I think you made a very small assumption error over here again so before we solve this I would like to discuss a small theory with you because it is directly related to this type of problem so whenever you have been provided the x y and z intercepts let me call them as a b c respectively which is made by a plane on the coordinate axis is and you've been asked the equation of this plane okay now this question is directly linked to this concept that's why let me take this small bit of theory first so when you've been given x intercept indirectly you have been given the coordinates of a which is nothing but a 0 0 when you have been given y intercept you have been given the coordinates of b which is 0 b 0 and you've been given the coordinates of c 0 0 c correct so when you know three points we have already discussed how to find the equation of a plane so when three points are known we just use the formula x minus x 1 y minus y 1 z minus z 1 x 2 minus x 1 y 2 minus y 1 z 2 minus z 1 x 3 minus x 1 y 3 minus y 1 z 3 minus z 1 equal to 0 correct so here you can take any one of the points let me take this point as my x 1 y 1 z 1 this has x 2 y 2 z 2 and this point as x 3 y 3 z 3 okay so equation would be x minus a y minus 0 z minus 0 x 2 minus x 1 will be minus a b 0 x 3 minus x 1 would be minus a again 0 and c okay so just expand this determinant and your equation will be there in front of you so it's x minus a times b c minus y times minus a c and z times plus a b okay so x minus a b c plus y a c plus z a b equal to 0 okay can write it as x b c y a c plus z a b equal to a b c now divide both sides by a b c okay so divide both sides by a b c so you get x by a y by b z by c equal to 1 now have you seen this kind of equation before in class 11 in 2d straight lines intercept form intercept form so this is the intercept form of the equation of a plane okay you will find a very stark resemblance between the line equation in 2d and plane equation in 3d okay a lot of concepts you'll find that the formula is almost the same it's just that in planes you have an extra dimension added to it so a line in 2d is nothing but a plane which is parallel to the y axis you can think like that so you're dealing with the projection of the plane on the x y plane is that fine now how is it helping us to solve the present question so please make a note of this this is important so how are we making use of this in our problem solving here now we have been given that centroid of this triangle is 1 2 4 centroid means a plus 0 plus 0 by 3 comma 0 plus b plus 0 by 3 comma 0 plus 0 plus c by 3 so this coordinate is given to us and what is that 1 2 4 so this is 1 this is 2 and this is 4 am I right which clearly means a by 3 is 1 so a is 3 okay b by 3 is 2 so b is 6 and c by 3 is 4 that means c is 12 so I can say that the equation is x by 3 y by 6 z by 12 is equal to 1 that becomes your desired equation of the plane if you want you can multiply throughout with a 12 and write down the answer is this clear now just a formula or generalization of the same concept so if you have been given that there is a plane there is a plane which cuts the coordinate axis is at let's say a b and c points okay says that the coordinate of the centroid here g is p q r then the equation of a plane can be written as x by 3 p plus y by 3 q plus z by 3 r equal to 1 just a generalization no need to remember this okay just a generalization of whatever I did in the given problem is that clear guys everyone okay so let me move on to the next problem okay so I'll move on to the next concept here that is the angle between the planes angle between two planes angle between two planes see this is a very simple concept so if you have been given two planes let's say white plane let me name it as pi 1 plane and a yellow plane let me call it as pi 2 plane okay and I have been asked to find the angle between these two planes the angle between these two planes means I have been asked to find out this angle okay so this angle here I have been asked to find out let me call it as theta okay now let's say the equation of these two planes are known to you let's say one is r is equal or r dot dot n1 plus d1 equal to 0 and other is r dot n2 plus d2 equal to 0 how do I find the angle between these two planes any suggestions for everyone let's say I have been given a1x plus b1y plus c1z plus d1 equal to 0 and a2x plus b2y plus c2z plus d2 equal to 0 yeah how do I find the angle between them the angle between normal vectors absolutely absolutely very good so when I say the angle between the two planes it's basically the angle between their respective normals so let's say this is your normal two plane pi 1 let me call it as n1 vector this is normal to your plane pi 2 let me call n2 vector so do you realize that this angle will also be theta right everybody agrees to this so the angle between them let's say I want my acute angle will be nothing but n1 dot n2 mod by mod n1 mod n2 okay in terms of the Cartesian form if you have been given the equation it just becomes mod a1 a2 plus b1 b2 very similar to the formula that we had for angle between two lines as well so nothing very different here just for formality okay again needless to say that when your planes are perpendicular theta has to be 90 degrees remember here theta is acute in case I forget yeah so if theta is 90 a1 a2 plus b1 b2 plus c1 c2 has to be zero and if the planes are parallel okay theta is zero degree that means a1 by a2 is equal to b1 by b2 is equal to c1 by c2 okay already known to you so nothing like new thing that I'm doing over here