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Question

A plane passing through the intersection of 2 planes ¯r.¯n1=d1 and ¯r.¯n2=d2 can be given by
¯r.(¯n1+λ¯n1)=d1+λd2

A
True
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B
False
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Solution

The correct option is A True
Let take the case of 2 planes meeting in a line. But we will see it will satisfy for other cases too.
Plane 1 will be ¯r.¯n1=d1
Plane 2 will be ¯r.¯n2=d2
We can write the plane through intersection as (¯r.¯n1d1)+λ(¯r.¯n2d2)=0. If we simplify this the form it takes is ¯r.(¯n1+λ¯n2)=d1+λd2. This can be viewed as ¯r.¯n3=d3, which represents the equation of a plane and satisfies for the points on the given planes only when they exist on the line of intersection unless λ=0. Therefore given statement is true.

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