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in Three-dimensional geometry by (36.3k points)

π1 and π2 are two planes passing through the origin. L1 and L2 are two lines passing through the origin such that L1 lies on π1 not on π2 and L2 lies on p2 not on π1. Show that there exist three points A, B, C whose permutation A', B', C' can be choosen such that

(a) A is on L1, B on π1 but not on L1 and C not on π1.

(b) A' is on L2, B' on π2 but not on L2 and C' not on π2.

1 Answer

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Best answer

We take A = A' as the origin and B = B' = any point other than A on the line intersection of π1 and π2.

Now, consider C = C' = any point niether on π1 nor π2.

Thus, in this case, both the conditions of (a) and (b) are fullfilled.

Similarly if we take,

A = non-origin point on L1

B = non-origin point on the line of intersection of π1 and π2 and

C = non-origin point on L2.

If we take A = C'', B = B', C = A', both the conditions of (a) and (b) are fulfilled.

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