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.