Consider the five points (0, 0, 0), (1, 0, 1), (2, 3, 2), (5, 4, 4), (7, 7, 7).

Question

Consider the five points (0, 0, 0), (1, 0, 1), (2, 3, 2), (5, 4, 4), (7, 7, 7).

Label the points A through E from left to right.

Find a plane that contains at least four of these points. Do all five of them lie on a common plane?

I can probably figure this one out by taking a bunch of cross products and looking for parallel ones or something, but I feel like that would be very time consuming if I don't take the cross products of the RIGHT points.

What is an efficient/systematic way to approach this problem on a test?

Explanation / Answer

All five points do NOT lie in a common plane. However, the four points (0, 0, 0), (1, 0, 1), (2, 3, 2), (7, 7, 7) DO lie in a common plane. The equation of the plane is z=x

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