Three Dimensional Geometry

Find the distance between the points A(1, 2, 3) and B(4, 6, 8).

The direction ratios of a line are proportional to (2, -3, 6). Which of the following can be the direction cosines of the line?

Find the equation of the plane passing through point P(1, 0, -1) and perpendicular to the vector n = 3i - 2j + 4k.

Find the angle between the lines whose direction ratios are (1, 2, 2) and (2, -1, 2).

Find the shortest distance from point Q(2, -1, 3) to the line passing through A(1, 0, 2) with direction vector d = 3i + 4j + 0k.

Find the coordinates of the foot of the perpendicular drawn from point P(3, 4, 5) to the plane 2x - y + 2z - 3 = 0.

Find the equation of the line of intersection of the planes x + y + z = 1 and 2x - y + 3z = 4.

Find the volume of the tetrahedron formed by the planes x=0, y=0, z=0 and 2x + 3y + 6z = 12.

Find the equation of the sphere with center at (2, -1, 3) and passing through the point (5, 2, 7).

If the line passes through points A(1, 2, 3) and B(4, 5, 6), find the coordinates of a point C on the line such that AC : CB = 2 : 1.