A few basic facts about planes in R3
- Every plane in R3 can be represented by an equation
ax + by + cz = d
(sometimes referred to as the linear equation of the plane), where at least one of the coefficients a, b, or c is nonzero. The nonzero vector <a, b, c> is normal -- i.e., perpendicular -- to the plane (refer to Section 12.5 in Stewart's Calculus, Early Transcendentals, 4th Edition).
- To see whether a point is on the plane or not, check whether the plane's equation becomes a true equality after the point's coordinates are substituted.
For example, the point (1,3,-2) is on the plane 7x - 4y - 5z = 5 [since (7)(1) - (4)(3) - (5)(-2) = 5], while the point (0,1,2) is not [since (7)(0) - (4)(1) - (5)(2) = -14 does not equal 5].
In other words the point (s,t,u) is on the plane ax + by + cz = d whenever x = s, y = t, z = u is a solution of the equation ax + by + cz = d.
- If one of the coefficients a, b, or c is zero, then the plane is parallel to the axis corresponding to the associated variable.
For example, the equation
4y - 5z = 2
represents a plane parallel to the x-axis. (This is a special case of a cylinder in R3 - refer to Section 12.6 in Stewart's Calculus, Early Transcendentals, 4th Edition).
- When two of the coefficients a, b, or c are zero, the plane is parallel to the two corresponding axes (see item 3), thereby making it parallel to the entire coordinate plane.
For example, the equation
x = -3
is associated with a plane parallel to the yz-plane (positioned three units away from it).
- The common part of two planes is the set of all points (x, y, z) that satisfy the simultaneous system of equations:
The planes are parallel if <a1, b1, c1> = k <a2, b2, c2> for some scalar k. If d1 = k d2, then the two equations represent the same plane, otherwise their common part is empty.
||a1 x + b1 y + c1 z = d1
||a2 x + b2 y + c2 z = d2
If the planes are not parallel, they intersect at a straight line.
- By adding p times each side of an equation of one plane and q times the corresponding side of an equation of another plane:
p ( a1 x + b1 y + c1 z ) = pd1
q ( a2 x + b2 y + c2 z ) = qd2
we obtain the combined equation:
(pa1+qa2)x + (pb1+qb2)y + (pc1+qc2)z = pd1+qd2,
which generally also represents a plane (unless the left hand side coefficients all become zero).
If the two planes intersect along the line L, then their "combined plane" also contains the entire line L. In this case, any plane containing L can be obtained by choosing the appropriate values for p and q. (Some results related to the concepts of vector spaces and bases are needed to justify this.) Therefore, geometrically, varying p and/or q will correspond to rotating the resulting plane around the line L as the axis of rotation.