Equation of a plane ( in space)		Equation of a line ( in space)
General form of the plane:  A x + By +Cz +D=0.  The normal vector of the plane:  n= Ai+ Bj+ Ck.  If any one of the A, B and C equals to 0,  then the corresponding axis is parallel to the plane.		The intersection of two planes:   Let A1x + B1y +C1z +D1=0, and  A2x + B2y +C2z +D2=0 be two given planes. Let n1 and n2 be the normal vectors of these two planes, then the direction vector of the line given by the intersection of the two planes is given by  v=n1 x n2 =.
Point-Normal form of the plane  is given by :  A (x-x0) + B (y-y0) +C (z-z0) +D=0.  P(x0, y0, z0) is a point in the plane with the normal vector  n = Ai+ Bj+ Ck.		Symmetric (Standard )Form of  a line:, where  P(x0, y0, z0) is a point in the line, and  v = l i+ m j+ n k. is the direction vector of the line.
3-point form of the plane is given by:  ,  where P(x1, y1, z1), Q(x2, y2, z2) and  R(x3, y3, z3)  are 3 points lying in the plane.		2-point form of the line is given by:,  where P(x1, y1, z1), Q(x2, y2, z2) are  2 points  in the line.
Interception form of a plane is given by:  ,  where a, b and c are the interceptions of x, y, z-axii respectively, i.e. the plane passes through the point (a, 0, 0), (0, b, 0) and (0, 0, c).		Parametric form of the line is given by:  x =x0+l t, y=y0+m t,  z=z0+n t,  where P0(x0, y0, z0) is a point in the line, and v = l i+ m j+ n k. is the direction vector of the line.  In vector form, OP(t)= OP0+tv, t is real.

Relationship between two planes:

• Let A₁ x + B₁ y +C₁ z +D₁=0, and A₂ x + B₂ y +C₂ z +D₂=0 be the equations of two planes.
• They are parallel if and only if   A₁ / A₂ = B₁ / B₂ = C₁ / C₂, i.e. their normal vector are parallel.
• They are perpendicular to each if and only if  their normal vectors are perpendicular,

i.e. A₁  A₂ + B₁  B₂ + C₁  C₂=0;
• The cosine angle between these two planes is given by cos  =.

Let A x + By +Cz +D=0 and  be the equations of  the plane and a straight line.
• The line is parallel to the plane if and only if A l +B m+C n=0.
• The line is normal to the plane if and only if A/l =B/m=C/n.
• The sine angle between a line and the plane is given by

sin  =.

Let and  be  the equations of two lines.
• The lines are parallel if and only if l₁/ l₂ =m₁/m₂=n₁/n₂.
• The lines are perpendicular if and only if  l₁l₂+m₁m₂+n₁n₂=0.
The cosine angle of these two lines is given by.
• The lines in space are called skew if and only if they are not parallel and nonintersecting .
• They are skew if and only if 

Note that shortest distance between two skew lines are well-defined.
• Two lines are not skew if they are parallel or they intersect each other.

• The distance from a point P(x₀, y₀, z₀) to a plane Ax +By +Cz+D=0 is given by .
• The distance from a point P(x₁, y₁, z₁) to the line is given by the length of the vector .

• Given the normal vector n and a point P₀(x₀, y₀, z₀)  ( as in fig. 1) below , one can easily write down the equation of the plane as < (x-x₀, y-y₀, z-z₀) , n >=0.

figure 1

figure 2

• Given three points P, Q and R in space ( as in figure 2 above), one can write down the equation of the plane passing through  P, Q and R, as follows:
• use the cross product of vectors PQ and PR to find a normal vector n of the plane,
• write down the quation by means of point-normal form.
• If the plane passes through the line l of common intersection of two other planes defined by f(x, y, z)=A₁x + B₁ y +C₁ z +D₁=0, and  g(x, y, z)=A₂ x + B₂ y +C₂ z +D₂=0. In general, the equation of the plane containing l (fig 3) is given by

 A g(x, y, z) +B f(x, y, z)=0, where A and B are two arbitrary constants.

figure 3

figure 4

•  Equation of a line:

Suppose two point Q(x₁, y₁, z₁) and R(x₂, y₂, z₂) of the line l' are given. (fig. 4) Let P(x, y, z) be a variable point on the line l', then these vectors PQ, RQ  are parallel, and hence we have .
If we set the common value of these ratios to be t, as a parameter to indicate various point, we can write the parametric form  x = x₁+l₁ t,  y = y₁+m₁ t, and  z = z₁+n₁ t, where l₁ =x₂ -x₁, m₁ =y₂ -y₁ and  n₁ =z₂ -z₁. Using vector notation, one can also write the parametric form as the convex combination: OP(t)=t OQ+(1-t) OR, where t is any real number.

• Intersection of  a line and a plane

Given a straight line l₁ :  and a plane Ax+ By+Cz+D=0.
An intersection point is given by  common coordinates. Let the point of the intersection is given by:  x = x₀+l t,  y = y₀+m t,  z = z₀+n t, for some particular t to be determined. Putting these into the equation of the plane, we have
A(x₀+l t )+B( y₀+m t )+ C( z₀+n t) +D=0, i.e. Ax₀+B y₀+ Cz₀+D+(Al+Bm+Cn) t =0.
• If Al+Bm+Cn is not zero, then there is unique soution for t, and hence the unique point of intersection.
• If Al+Bm+Cn=0 and Ax₀+B y₀+ Cz₀+D=0 (i.e.P(x₀, y₀, z₀) lies on the plane), then all  the point on the line lies in the plane, so the plane contains the line l.
• If Al+Bm+Cn=0 and Ax₀+B y₀+ Cz₀+D is not zero, there is no solution for t, and hence no intersection.

• Intersection of two lines:
• Let and  be two given lines, they intersects if and only if there exist x = x₁+l₁ t=x₂+l₂ s,  y = y₁+m₁ t= y₂+m₂ s,  z = z₁+n₁ t= z₂+n₂ s, for some particular t, s to be determined.  So the  linear homogeneous system has a non-trivial solution  ( t, -s, 1), i.e. D=det=0, which is a necessary condition.
• If D is not zero, then the lines are skew ones.
• If D=0, then we can solve the solutions t, s, and determine the point(s) of intersection. It should be noted that there may be an unique intersection point or infinitely many intersection points according to the rank of the 3x3 matrix is of rank= 2 or 1 respectively. In the latter case, these two lines coincide and hence there are infinitely many points of intersection.
• Intersection of 2 planes:

Let A₁ x + B₁ y +C₁ z +D₁=0, and A₂ x + B₂ y +C₂ z +D₂=0 be the equations of two planes.
let P(x, y, z) be any point of  the common intersection of these planes, then (x, y, z) is the solution of the following  imhomogeneous linear system: [S |T]=. In general the solution set depends on the row echelon form of the matrix.
• If  rank S =2, then rank [S | T]=2, then the planes intersect at a line.
• If rank S=1, then rank[ S | T]=2, then the planes do not intersect, and they are parallel.
• If rank S= rank[S | T] =1, then the planes coincide.
Recall that the rank of a m x n matrix A is defined to be number of leading 1's in the row-echelon form of A.