x – x1 / a = y – y1 / b = z – z1 / c = – 2 ( ax1 + by1 + cz1 + d ) / a 2 + b 2 + c 2
13 . The foot ( x , y , z ) of a point ( x1 , y1 , z1 ) in a plane ax + by + cz + d = 0 is given by x – x1 / a = y – y1 / b = z – z1 / c = – ( ax1 + by1 + cz1 + d ) / a 2 + b 2 + c 2
14 . Since , x , y and z-axes pass through the origin and have direction cosines ( 1 , 0 , 0 ), ( 0 , 1 , 0 ) and ( 0 , 0 , 1 ), respectively . Therefore , their equations are
x – axis : x – 0 / 1 = y – 0 / 0 = z – 0 / 0 y – axis : x – 0 / 0 = y – 0 / 1 = z – 0 / 0 z – axis : x – 0 / 0 = y – 0 / 0 = z – 0 / 1
Plane
A plane is a surface such that , if two points are taken on it , a straight line joining them lies wholly in the surface .
General Equation of the Plane
The general equation of the first degree in x , y , z always represents a plane . Hence , the general equation of the plane is ax + by + cz + d = 0 . The coefficient of x , y and z in the cartesian equation of a plane are the direction ratios of normal to the plane .
Equation of the Plane Passing Through a Fixed Point
The equation of a plane passing through a given point ( x1 , y1 , z1 ) is given by a ( x – x1 ) + b ( y — y1 ) + c ( z — z1 ) = 0 .
Normal Form of the Equation of Plane
( i ) The equation of a plane , which is at a distance p from origin and the direction cosines of the normal from the origin to the plane are l , m , n is given by lx + my + nz = p .