Question

Equation of plane passing through intersection of two planes and perpendicular to a line

Answer 1

Two planes can intersect in the three-dimensional space. Imagine two adjacent pages of a book. These two pages are nothing but an intersection of planes, intersecting each other and the line between them is called the line of intersection. A new plane i.e. a third plane can be given to be passing through this line of intersection of planes. We are to find out the equation of this plane.

Let us assume that the equation of the first plane is π1 and that of the second is π2. The equation of our required plane is π and we are to find out this equation itself. This equation is given by –

π1+λπ2=0………..(1)

Now, the equation of the planes can be given in vector form or Cartesian form. We shall explore both these forms in the following sections and see how the equation of the required plane can be found using the given information.

Vector form

If the equation of the two planes is given in Vector form –

(r⃗ –a1→).n1→=0

and

(r⃗ –a2→).n2→=0

These two equations can alternatively be written as –

r⃗ .n1→–d1→=0

and

r⃗ .n2→–d2→=0

So the equation of the required plane by using (1) can be written as –

(r⃗ .n1→–d1→)+λ(r⃗ .n2→–d2→)=0………..(2)

i.e.

r⃗ (n1→+λn2→)–(d1→+λd2→)

By substitution, we have –

(xi^+yj^+zk^).(n1→+λn2→)–(d1→+λd2→)

Usually a point on the third plane will be given to you. You must then substitute the coordinates of the point for x, y and z to find the value of λ. Then use this value of λ to get the equation of the plane from (2). This equation will be nothing but the equation of the required plane that is passing through the line of intersection of two planes and a given point.