Thread: Problem help - Orthogonal Bases

Consider the following Subspace on R3:

U={(x1,x2,x3) € R3: x1-2x2+x3=0}

Which of the following group of vectors is an orthogonal basis of U?

A - {(1,-2,1), (-1,0,1)}
B - {(-1,0,1), (1,1,1)}
C - {(2,1,0), (-1,0,1)}
D - {(2,1,0), (0,0,1)}
E - {(1,-2,1), (-1,0,1), (1,1,1)}

Can anyone give me a detailed resolution for this exercise?

First note that the equation describes a plane in R^3. Specifically, recall that a plane may be specified by the set of all points whose displacement vectors from a proprietary point are normal to a specific vector, called the normal vector. Normality is defined by the dot product vanishing, so note that your equation may be rewritten as A normal vector to your plane is thus (1, -2, 1).
The two basis vectors must have a dot product of 0 to be orthogonal and must have a cross product that is a scalar multiple of the normal vector, since they must lie in that plane.