 Hello and welcome to the session. In this session, first we are going to discuss reduction of general equation of the plane to the normal form. As we have discussed, general equation of the plane is given by A x plus B y plus C z plus D is equal to 0 where A B C D are constants and normal form of the plane is given by L x plus M y plus M z minus P is equal to 0 where L M N are the direction cosines of the normal and P is the length of the normal from origin to the plane. Now we shall discuss reduction of general equation to normal form. Let the general equation of the plane B plus B y plus C z plus D is equal to 0, mark this equation as 1. The equation of the same plane in the normal form is L x plus M y plus M z minus P is equal to 0, mark this equation as 2. Where P is always positive, now equation 1 and equation 2 are identical if L upon A is equal to M upon B is equal to M upon C is equal to minus P upon D which implies that minus P upon D is equal to L upon A is equal to M upon B is equal to M upon C. As we know that if A upon B is equal to C upon D is equal to E upon F then E is equal to plus minus square root of A squared plus C squared plus E squared hold upon square root of B squared plus D squared plus M squared which implies that minus P upon D is equal to plus minus square root of L squared plus M squared plus N squared upon square root of A squared plus B squared plus C squared and we know that L squared plus M squared plus M squared is equal to 1. Therefore we have minus P upon D is equal to plus minus 1 upon square root of A squared plus B squared plus C squared. Now as minus P upon B is equal to plus minus 1 upon square root of A squared plus B squared plus C squared and P is always positive, now we will take positive or negative sign accordingly as B is negative or positive, if D is positive then L is equal to minus A upon square root of A squared plus B squared plus C squared and we can also write minus A upon square root of summation of A squared M is equal to minus B upon square root of A squared plus B squared plus C squared or we can also write minus B upon square root of summation of A squared M is equal to minus C upon square root of A squared plus B squared plus C squared or we can also write minus C upon square root of summation of A squared and P is given by B upon square root of summation of A squared If D is negative then L is equal to A upon square root of summation of A squared M is equal to B upon square root of summation of A squared N is equal to C upon square root of summation of A squared E is given by minus D upon square root of summation of A squared hence the normal form of plus B y plus C z plus D is equal to 0 is given by minus A upon square root of summation of A squared X minus B upon square root of summation of A squared Y minus C upon square root of summation of A squared Z is equal to D upon square root of summation of A squared when D is positive A upon square root of summation of A squared X plus B upon square root of summation of A squared Y plus C upon square root of summation of A squared Z is equal to minus D upon square root of summation of A squared when D is negative Now we are going to discuss equations of by sectors of an angle between two planes Let us suppose equations of the two planes B 1 X plus B 1 Y plus C 1 Z plus D 1 is equal to 0 mark this equation as 1 and A 2 X plus B 2 Y plus C 2 Z plus D 2 is equal to 0 mark this equation as 2 Let P with the coordinates X Y Z be any point on any one of the planes by second the angles between the planes 1 and 2 then the length of the perpendicular from P with coordinates X Y Z to the planes 1 and 2 must be numerically equal The equations of the by sectors of the angles between the planes 1 and 2 plus B 1 Y plus C 1 Z plus D 1 upon square root of A 1 square plus B 1 square plus C 1 square is equal to plus minus A 2 X plus B 2 Y plus C 2 Z plus D 2 upon square root of A 2 square plus B 2 square plus C 2 square one of these two planes by 6 3 acute angle and the other the obtuse angle between the planes 1 and 2 Let us take an example find the equations of the by sector planes of the angles between the planes 3 X minus 2 Y plus 6 Z plus 4 is equal to 0 and 2 X minus Y plus 2 Z plus 3 is equal to 0 Now we are given two planes that is 3 X minus 2 Y plus 6 Z plus 4 is equal to 0 mark this equation as 1 and 2 X minus Y plus 2 Z plus 3 is equal to 0 mark this equation as 2 Now if P with the coordinates X Y Z be any point on the by sector planes the perpendicular from P with the coordinates X Y Z to the two planes must be numerically equal that is We have 3 X minus 2 Y plus 6 Z plus 4 upon square root of 3 square plus minus 2 the whole square plus 6 square is equal to plus minus 2 X minus Y plus 2 Z plus 3 upon square root of 2 square plus 2 Z plus 3 is equal to 0 minus 1 the whole square plus 2 square which implies that 3 X minus 2 Y plus 6 Z plus 4 upon square root of 3 square that is 9 plus of minus 2 the whole square that is 4 plus 6 square that is 36 So we have 9 plus 4 plus 36 that is 39 is equal to plus minus 2 X minus Y plus 2 Z plus 3 upon square root of 2 square that is 4 plus of minus 1 the whole square that is 1 plus 2 square that is 4 So we have square root of 4 plus 1 plus 4 that is square root of 9 therefore we have 3 X minus 2 Y plus 6 Z plus 4 upon 7 is equal to plus minus 2 X minus Y plus 2 Z plus 3 upon square root of 9 that is 3 Mark this equation as 3 which are the equations with 2 bisector planes taking positive sign in equation 3 we have 3 X minus 2 Y plus 6 Z plus 4 by 7 is equal to 2 X minus Y plus 2 Z plus 3 by 3 On cross multiplication we get 3 into 3 X that is 9 X minus of 3 into 2 Y that is 6 Y plus 3 into 6 Z that is 18 Z plus 3 into 4 that is 12 is equal to 7 into 2 X that is 14 X minus of 7 into Y is 7 Y plus 7 into 2 Z is 14 Z plus 7 into 3 is 21 Now taking all the terms from right hand side we get 14 X minus 7 Y plus 14 Z plus 21 minus 9 X plus 6 Y minus 18 Z minus 12 is equal to 0 On solving further we get 5 X minus Y minus 4 Z plus 9 is equal to 0 Now again taking negative sign in equation 3 we get 3 X minus 2 Y plus 6 Z plus 4 upon 7 is equal to minus of 2 X minus Y plus 2 Z plus 3 upon 3 On cross multiplication we get 3 into 3 X that is 9 X minus of 3 into 2 Y that is 6 Y plus 3 into 6 Z that is 18 Z plus 3 into 4 that is 12 is equal to minus of 7 into 2 X that is 14 X minus of 7 into Y is 7 Y plus 7 into 2 Z is 14 Z plus 7 into 3 is 21 which again implies that 9 X minus 6 Y plus 18 Z plus 12 is equal to minus of 14 X plus 7 Y minus of 14 Z minus of 21 Taking all the terms on the left hand side of the equation we get 9 X minus 6 Y plus 18 Z plus 12 plus 14 X minus 7 Y plus 14 Z plus 21 is equal to 0 Therefore we get 23 X minus 13 Y plus 32 Z plus 33 is equal to 0 Therefore the equations of the disector planes are 5 X minus Y minus 4 Z plus 9 is equal to 0 and 23 X minus 13 Y plus 32 Z plus 33 is equal to 0 This completes our session. Hope you enjoyed this session.