How would I know that the orthonormal set of the plane x+y+2z=0 is a basis for the plane.?

Answer 1

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Answer 2

Planes are 2-dimensional. If you have 2 vectors on the plane and their cross product is not zero, then they form a basis of your plane. Since you have an orthonormal set (of 2 vectors) , the vectors in that set are perpendicular. In particular, their cross product has magnitude 1 which is not zero. This is definition chasing.

Answer 3

If adding and subtracting multiples of the vectors making up the alleged basis for the plane can reach every point in the plane, then it is a basis.