To check if two planes have a line in common is quite simple, but how do I determine if two planes have a point in common?
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Write the equation of the planes as $ax+by+cz=d$ where $a^2+b^2+c^2=1$ so that $d $ is the distance of the plane from the origin in $\mathbb R^3$. The only case when they don't intersect is when they are skew planes, parallel but at different distance from the origin. Otherwise, they are either the same plane or their intersection is a line.
Two planes have just a point in common in spaces with dimension 4 or higher. Let's name the planes V2 and V'2, dimension "dim". Sign "_" will be conjunction of spaces (linear span of their two basis), sign "^" will be their intersection (which is also a space). Then according to theorem about "dimension of conjunction and intersection of spaces" we can write:
dim (V2_V'2)+dim(V2^V'2)=dim(V2)+dim(V'2)=2+2=4 .
If dim(V2_V'2)=4 then din(V2^V'2)=0 . That means that the dimension of the intersection of the spaces is 0 --> the intersection is just a point. The problem is that it is a little bit hard to imagine it. Have a good fun!