 Hello and welcome to this session. In this session we will discuss a question that says that reduce 2x plus y minus 2z plus 15 is equal to 0 to the normal form and determine the length and the angles that normal links with the axis, that is the direction angles. Now before starting the solution of this question, we should know our result. And that is the method to reduce the general equation of the plane to the normal form. Now let the general equation of a plane is Ax plus By plus Cz plus t is equal to 0. And the normal form of the given plane is Ax plus My plus nz minus p is equal to 0. Or you can write Ax plus My plus nz is equal to p. Now if in this equation d is positive then l is equal to minus a over square root of summation a square. m is equal to minus b over square root of summation a square equal to minus c over square root of summation a square. And if d is negative then only we have to change the signs of all of these. Plus By plus Cz plus d is equal to 0 a square into x minus b over square root of summation a square into y minus c over square root of summation a square into z is equal to d over summation a square. d is positive. And here this is the equation in the normal form that is lx plus my plus nz is equal to p where p is equal to d over square root of summation a square and these are the values of m. And this is the equation of the normal form of this plane when d is negative. And also where m is equal to cos alpha m is equal to cos beta and n is equal to cos gamma where alpha, beta and gamma are called the direction angles that is the angles which normal makes with the axis. And here summation a square is equal to a square plus b square plus c. So this result will work out as a key idea for solving this question. And now we will start with the solution. We have to reduce this equation to the normal form and also we have to determine the length plus y minus this result which is given in the key idea. We can reduce the equation of the given plane to the normal form. Now here b square plus c square is equal to square root of b is 1 and c is minus 2. So it will be equal to square root of 2 square plus 1 square plus of minus 2 square. This is equal to square root of 4 plus 1 plus 4 which is equal to root 9. Now let this be equation number 1 which is given by equation number 1 that is minus summation a square that is root line b that is 1 root line into y into z is equal to d which is 15th of summation a square which implies minus 2 over 3 into x minus 1 over 3 into y minus of minus 2 will be plus over 3 into z is equal to d into y. Now the equation of the plane is reduced into normal form that is equal to minus 2 by 3 m is equal to minus 1 by 3 and is equal to 2 by 3. Now we have l is equal to minus 2 by 3. Now we know that l is equal to cos alpha m is equal to cos beta and n is equal to cos gamma where alpha, beta and gamma are the direction angles minus 2 by 3 which further implies y3 equal to minus 1 by 3 implies beta is equal to minus 1 by 3 which implies is equal to cos inverse minus 1 by 3 implies is equal to cos inverse 2 by 3 length of the normal p is equal to 5 is 2 by 3 beta is minus 1 by 3. So this is the solution of the given equation and that's all for this session. Hope you all have enjoyed the session.