Let the equation of a plane be ax + by + cz + d = 0

1. When P and Q are on the same side of the plane

=> (ax

_{1}+ by_{1}+ cz_{1 }+ d)/ (ax_{2}+by_{2}+ cz_{2}+ d) < 0
=> (ax

_{1}+ by_{1}+ cz_{1 }+ d)/ (ax_{2}+by_{2}+ cz_{2}+ d) > 0
=> ax

_{1}+ by_{1}+ cz_{1 }+ d and ax_{2}+by_{2}+ cz_{2}+ d are of the same sign.
2. When P and Q are on the opposite side of the plane

=> (ax

_{1}+ by_{1}+ cz_{1 }+ d)/ (ax_{2}+by_{2}+ cz_{2}+ d) < 0
=> ax

_{1}+ by_{1}+cz_{1}+ d and ax_{2}+ by_{2}+ cz_{2}+ d are of opposite sign.**Perpendicular Distance of a point from a plane:**The length of the perpendicular from the point P (x

_{1}, y

_{1}, z

_{1}) on the plane ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d|/ √ (a₁² + b₁² + c₁²)

**Distance between Parallel Planes:**

Let ax

_{1}+ by_{1}+ cz_{1}+ d =0 … (i)
And, ax

_{2}+ by_{2}+ cz_{2}+ d =0 … (ii)
Be two parallel planes

Then, the distance between them |d₁ - d₂|/ √ (a₁²
+ b₁² + c₁²)

**Image of a Point in a Plane:**Let P an 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.

In order to find the image of a point in a given plane. We
may use the following algorithm

**Algorithm:**

**Step I:**Write the equations of the line passing through P and normal - to the given plane as (x - x₁)/ a = (y - y₁)/ b = (z - z₁)/ c.

**Step II:**Write the coordinates of image Q as (x

_{1}+ ar, y

_{1}+ br, z

_{1}+ cr).

**Step III:**Find the coordinates of the mid-point R of PQ.

**Step IV:**Obtain the value of r by putting the coordinates of R in the equation of the plane.

**Step V:**Put the value of r in the coordinates of Q.

**Example:**Find the image of the point (3, -2, 1) in the plane 3x – y + 4z = 2.

**Solution:**Let Q be the image of the point P (3, -2, 1) in the plane 3x – y + 4z = 2. Then, PQ is normal to the plane.

Therefore, direction ratios of PQ are 3, -1, 4. Since PQ
passes through P (3, -2, 1) and has direction ratios 3, -1, 4.