## What is a Plane in 3D?

### Definition of plane and its equations

If point P(x, y, z) moves according to certain rule, then it may lie in a 3-D region on a surface or on a line or it may simply be a point. Whatever we get, as the region of P after applying the rule, is called locus of P. Let us discuss about the plane or curved surface. If Q be any other point on it’s locus and all points of the straight line PQ lie on it, it is a plane. In other words if the straight line PQ, however small and in whatever direction it may be, lies completely on the locus, it is a plane, otherwise any curved surface.

**General equation of plane:**Every equation of first degree of the form Ax + By + Cz + D = 0 represents the equation of a plane. The coefficients of x, y and z i.e., A, B, C are the direction ratios of the normal to the plane.**Equation of co-ordinate planes:**XOY-plane : z = 0, YOZ -plane : x = 0, ZOX-plane : y = 0.**Equation of plane in various forms:****Intercept form:**If the plane cuts the intercepts of length a, b, c on co-ordinate axes, then its equation is

.**Normal form:**Normal form of the equation of plane is , where l, m, n are the d.c.’s of the normal to the plane and p is the length of perpendicular from the origin.

**Equation of plane in particular cases:**

Equation of plane through the origin is given by Ax + By + Cz = 0.

i.e, if D = 0, then the plane passes through the origin.**Equation of plane parallel to co-ordinate planes or perpendicular to co-ordinate axes:**- Equation of plane parallel to YOZ-plane (or perpendicular to x-axis) and at a distance ‘a’ from it is x = a.
- Equation of plane parallel to ZOX-plane (or perpendicular to y-axis) and at a distance ‘b’ from it is y = b.
- Equation of plane parallel to XOY-plane (or perpendicular to z-axis) and at a distance ‘c’ from it is z = c.

**Equation of plane perpendicular to co-ordinate planes or parallel to co-ordinate axes:**- Equation of plane perpendicular to YOZ-plane or parallel to x-axis is By + Cz + D = 0.
- Equation of plane perpendicular to ZOX-plane or parallel to y-axis is Ax + Cz + D = 0.
- Equation of plane perpendicular to XOY-plane or parallel to z-axis is Ax + By + D = 0.

**Equation of plane parallel to a given plane:**

Plane parallel to a given plane ax + by + cz + d = 0 is ax + by + cz + d’ = 0,*i.e.*only constant term is changed.**Equation of plane passing through the intersection of two planes:**

Equation of plane through the intersection of two planes P = a_{1}x + b_{1}y + c_{1}z + d_{1}= 0 and Q = a_{2}x + b_{2}y + c_{2}z + d_{2}= 0 is P + λQ = 0, where λ is the parameter.

### Equation of plane passing through the given point

**Equation of plane passing through a given point:**

Equation of plane passing through the point (x_{1}, y_{1}, z_{1}) is A(x – x_{1}) + B(y – y_{1}) + C(z – z_{1}) = 0, where A, B and C are d.r.’s of normal to the plane.**Equation of plane through three points:**

The equation of plane passing through three non-collinear points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is

### Foot of perpendicular from a point A(α, β, γ) to a given plane ax + by + cz + d = 0.

If AP be the perpendicular from A to the given plane, then it is parallel to the normal, so that its equation is

Any point P on it is (ar + α, br + β, cr + γ). It lies on the given plane and we find the value of r and hence the point P.

**Perpendicular distance:**

The length of the perpendicular from the point P(x_{1}, y_{1}, z_{1}) to the plane ax + by + cz + d = 0 is

Distance between two parallel planes Ax + By + Cz + D_{1}= 0 and Ax + By + Cz + D_{2}= 0 is

**Position of two points w.r.t. a plane:**

Two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) lie on the same or opposite sides of a plane ax + by + cz + d = 0 according to a_{1}x + b_{1}y + c_{1}z + d and a_{2}x + b_{2}y + c_{2}z + d are of same or opposite signs. The plane divides the line joining the points*P*and*Q*externally or internally according to*P*and*Q*are lying on same or opposite sides of the plane.

### Angle between two planes

Angle between the planes is defined as angle between normals to the planes drawn from any point. Angle between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 is

### Equation of planes bisecting angle between two given planes

Equations of planes bisecting angles between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 are

(i) If angle between bisector plane and one of the plane is less than 45o, then it is acute angle bisector, otherwise it is obtuse angle bisector.

(ii) If a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} is negative, then origin lies in the acute angle between the given planes provided d_{1} and d_{2} are of same sign and if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} is positive, then origin lies in the obtuse angle between the given planes.

### Image of a point in a plane

Let P and Q be two points and let π be a plane such that

(i) Line PQ is perpendicular to the plane π, and

(ii) Mid-point of PQ lies on the plane π.

Then either of the point is the image of the other in the plane π.

To find the image of a point in a given plane, we proceed as follows

(i) Write the equations of the line passing through P and normal to the given plane as

(ii) Write the co-ordinates of image Q as (x_{1} + ar, y_{1} + br, x_{1} +cr).

(iii) Find the co-ordinates of the mid-point R of PQ.

(iv) Obtain the value of r by putting the co-ordinates of R in the equation of the plane.

(v) Put the value of r in the co-ordinates of Q.

### Coplanar lines

Lines are said to be coplanar if they lie in the same plane or a plane can be made to pass through them.

Condition for the lines to be coplanar: