Three Dimensional Geometry
Find the equation of the plane through the points (2, 1, 1) and (1, 3, 4) and perpendicular to the plane x 2y+4z=10.
Show that the equation ax+by+d=0 represents a plane parallel to the z-axis. Hence, find the equation of a plane which is parallel to the z-axis and passes through the points A(2,−3,1) and B(−4,7,6).
Find the vector equation of the plane passing through the point (1,1,1) and parallel to the plane r⋅(2i^−j^+2k^)=5.
Show that the distance of the point of intersection of the line 3x−2=4y+1=12z−12 and the plane x−y+z=5 from the point (−1,−5,−10) is 13 units.
Find the equation of the plane passing through the intresection of the planes x−2y+z=1 and 2x+y+z=8 and parallel to the line with direction ratio proportional to 1,2,1, find also the perpendicular distance of (1,1,1) from this plane.
For the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin.