Example: 10  
Find the equation of the plane passing through the points and .

Solution: 10  
Let be any arbitrary point in the plane whose equation we wish to determine:
Since will be perpendicular to this plane, we must have
We could have proceeded alternatively as follows: using the result of the last example, any arbitrary plane through will be of the form
If this passes through and , we have
and
Thus the equation of the plane is

Example: 11  
Find the equation of the plane intercepting lengths , and on the ,  and axis respectively.

Solution: 11  
The plane passes through the points , and . For any variable point in this plane, we have (as discussed in the previous section),
This general equation has the same form as the equation of the line in intercept form; which further proves the analogy between the formulae in two and in three dimensions.

Example: 12  
A plane is at a distance from the origin and the direction cosines of the (outward) normal to it are , , . Find its equation.

Solution: 12  
The unit vector normal to the plane is
For any point in the plane, we have
This is the required equation; it is called the normal form of the plane’s equation. As an exercise, convert the general equation of the plane
into normal form.
