Definition of Plane
Let there be three non-collinear points. There exist 3 distinct lines, which pass through these points taken 2 at a time. The triangle so formed is a planar surface.
A line joining any 2 points on a plane always lies on the plane.
Equation of Plane : Normal Form
Consider a plane. Let be a point on this plane, such that. Position vector of , would be , if is the magnitude of . is the unique vector along . Let be any point on the plane. Clearly, .
Cartesian Equivalent of normal form is , where and are the direction ratios of any line normal to the plane.
Equation of Plane through a Point and Normal to a Vector
Let be the point and be the vector. If is perpendicular to the plane, it will be perpendicular to any vector in that plane. Let be a point on the plane. Then,
Cartesian Equivalent of above equation is
if and .
Equation of Plane through a Point and Parallel to 2 Non-parallel Non-zero Vectors
Let be the point and and be the vectors. will perpendicular to the plane, hence,
Equation of Plane through 3 Non-collinear Points
Let and be the points and be any point on the plane.Thus, , and will be coplanar. Thus,
This is the required equation.
Equation of Plane through Intersection of 2 Planes
This is similar to the equation of family of straight lines through a point (in 2D).
Note that intersection of 2 planes gives us a straight line and infinitely many planes can pass through the intersection of 2 planes. Let the planes be and . The vector equation of a plane through the intersection of these 2 is given by
Angle between Two Planes
Angle between 2 planes is the angle between their normals. So, if the normals are and , then the angle will be
For acute angle, we take the modulus of the RHS.
If is , planes are perpendicular. If is , planes are either parallel or coincident.
Angle between a Line and a Plane
Let the line be . Let be a vector normal to the plane. Clearly, angle between the line and the plane will be equal to minus the acute angle made by the line (or ) and .
When are 2 Lines Coplanar?
We know that in 3 dimensions, lines can be skew or coplanar. Let the lines be and be the lines. The lines will be coplanar if and only if the scalar triple product of the following vectors is zero :
Distance of a Point from a Plane
This is similar to distance of a point from a line. We find the value of parameter . Let be the point and be the plane. Let the foot of the perpendicular from on the plane be . Vector equation of line will be
Since lies on the plane, we have
From this, we get the value of . Having obtained , we can easily get coordinates of and eventually the distance .