Cartesian form: If the lines are

Then, shortest distance,

Distance between two Parallel Lines: If two lines l1 and l2 are parallel, then they are coplanar. Let the lines be r =a1+λb  and r =a2+μb, then the distance between parallel lines is

Note: If two lines are parallel, then they both have the same DR’s.

Distance between Two Points: The distance between two points P (x1, y1, z1) and Q (x2, y2, z2) is given by

Mid-point of a Line: The mid-point of a line joining points A (x1, y1, z1) and B (x2, y2, z2) is given by

Plane: A plane is a surface such that a line segment joining any two points of it lies wholly on it. A straight line which is perpendicular to every line lying on a plane is called a normal to the plane.

Equations of a Plane in Normal form
Vector form: The equation of plane in normal form is given by r .n =d, where n  is a vector which is normal to the plane.
Cartesian form: The equation of the plane is given by ax + by + cz = d, where a, b and c are the direction ratios of plane and d is the distance of the plane from origin.
Another equation of the plane is lx + my + nz = p, where l, m, and n are direction cosines of the perpendicular from origin and p is a distance of a plane from origin.
Note: If d is the distance from the origin and l, m and n are the direction cosines of the normal to the plane through the origin, then the foot of the perpendicular is (ld, md, nd).