Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, – 1), (4, 3, – 1).

Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Find the equation of the plane passing through the point (3,-3, 1) and perpendicular to the line joining the points (3, 4, - 1) and (2, - 1,5). Also find the coordinates of the foot of the perpendicular, the equation of the perpendicular line and the length of perpendicular drawn from the origin to the plane.

Determine the value of k so that the lie joining the points A(k,1,-1), B(2,0,2k) is perpendicular to the line joining the points C(4,2k,1) and D(2,3,2)

Find the equation of the plane passing through the point A(1, 2, 1) and perpendicular to the line joining the points P(1, 4, 2) and Q(2, 3, 5). Also, find the distance of this plane from the line (x+3)/(2)=(y-5)/(-1)=(z-7)/(-1)

Show that the line joining (2, -3) and (- 5,1) is : perpendicular to the line joining (4, 5) and (0, - 2).

Find the vector and cartesian equation of a line through the point (1, – 1, 1) and perpendicular to the lines joining the points (4, 3, 2), (1, -1,0) and (1, 2,-1), (2, 1, 1).

Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (– 1, – 2, 1), (1, 2, 5).

Find the equation of the plane passing through the points (1,2,1) and perpendicular to the line joining the points (1,4,2) and (2,3,5), Also, find the perpendicular distance of the plane from the origin.

Show that the line : through (2, - 3) and (-1, 2) is perpendicular to the line joining (3, 7) and (- 2, 4).

Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the plane determined by the points P (2, 1, 2), Q(3, 1, 0) and R (4, -2,1)