 Good morning friends. I am Purva and today we will discuss the following question. In the following find the coordinates of the foot of the perpendicular drawn from the origin. x plus y plus z is equal to 1. Let us begin with the solution now. Now we are given that the equation of the plane is x plus y plus z is equal to 1. Let the coordinates of the foot of the perpendicular of point P from the origin to the plane be x1, y1, z1. Then the direction ratios of line OPR, x1, y1, z1. Writing the equation of the plane in normal form we get x upon under root 3 plus y upon under root 3 plus z upon under root 3 is equal to 1 upon under root 3. We have got this equation by dividing the equation of the plane by under root 3 where this under root 3 is equal to under root of coefficient of x square that is 1 square plus coefficient of y square that is 1 square plus coefficient of z square that is 1 square. Now this equation is of the form lx plus my plus nz is equal to d which is the Cartesian equation of plane in normal form where we have l, m and n are the direction cosines of the normal and d is the distance of plane from origin. So comparing these two equations we get here 1 upon root 3, 1 upon root 3, 1 upon root 3 are the direction cosines of OP. Now since the direction cosines and the direction ratios of a line are proportional we have x1 upon 1 upon root 3 is equal to y1 upon 1 upon root 3 is equal to z1 upon 1 upon root 3 is equal to some k. Now here x1 by 1 and z1 are the direction ratios and 1 upon root 3, 1 upon root 3 and 1 upon root 3 are the direction cosines that is we get x1 is equal to k upon root 3, y1 is equal to k upon root 3 and z1 is equal to k upon root 3. We mark this as 1. Now substituting these values in the equation of the plane we get k upon root 3 plus k upon root 3 plus k upon root 3 is equal to 1. This implies 3k is equal to under root 3 which further implies k is equal to 1 upon root 3. Now putting this value of k in 1 we get x1 is equal to 1 upon root 3 into 1 upon root 3 which is equal to 1 upon 3, y1 is equal to 1 upon root 3 into 1 upon root 3 which is again equal to 1 upon 3 and z1 is equal to 1 upon root 3 into 1 upon root 3 which is again equal to 1 upon 3. Thus the coordinates of the foot of the perpendicular are 1 upon 3 comma 1 upon 3 comma 1 upon 3. So we have got our answer as 1 upon 3 comma 1 upon 3 comma 1 upon 3. Hope you have understood the solution. Bye and take care.