In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
a.2x+3y+4z-12=0
b.3y+4z-6=0
c. x+y+z=1
d. 5y+8=0

Find the coordinates of the foot of the perpendicular P from the origin to the plane 2x-3y+4z-6=0

In each of the following cases , determine the direction cosines of the normal to the plane and the distance from the origin . a. z=2 , b.x+y+z=1 , c. 2x+3y-z=5, d.5y+8=0

Find the coordinates of the foot of the perpendicular and length of the perpendicular from the point (4,3,2) to the plane x + 2y + 3z = 2.

Given the plane 5x-2y+4z-9=0 i.Find the coordinates of foot of the perpendicular from the origin to the plane. ii. Find the vector equation and the cartesian equation of this perpendicular .

The foot of the perpendicular from A (1, 0, 0) to the line (x-1)/2=(y+1)/(-3)=(z+10)/8 is

A plane passes through a fixed point (a ,b ,c)dot Show that the locus of the foot of the perpendicular to it from the origin is the sphere x^2+y^2+z^2-a x-b y-c z=0.

In the following cases, find the distance of each of the given points from the corresponding given plane. {:("Point","Plane"),("a. (0,0,0),3x-4y+12z=3),(b. (3,-2,1),2x-y+2z+3=0),(c.(2,3,-5),x+2y-2z=9),(d.(-6,0,0),2x-3y+6z-2=0):}

In the follwing cases , determine whether the given planes are parallel or perpendicular and in case they are neither , find the angles between them. a. 7x+5y+6z+30=0 and 3x-y-10z+4=0 b. 2x+y+3z-2=0 and x-2y+5=0 c. 2x-2y +4z +5=0 and 3x-3y+6z-1=0 d. 2x-y+3z-1=0 and 2x-y+3z+3=0 e. 4x+8y+z-8=0 and y+z-4=0

Find the vector and Cartesian equations of the plane passing through the point (1,1,-1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0