Find a basis for the subspace of vectors orthogonal to <1,2,1>.

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- Mar 2nd 2008, 12:59 AMchaddyBasis for a subspace
Find a basis for the subspace of vectors orthogonal to <1,2,1>.

- Mar 2nd 2008, 03:08 AMmr fantastic
Let the vectors orthogonal to <1, 2, 1> be <a, b, c>. Equating the dot product to zero gives a + 2b + c = 0 => a = -2b - c.

Then vectors orthogonal to <1, 2, 1> have the form <-2b - c, b, c>.

One possible basis is therefore the pair of vectors <-2, 1, 0> and <-1, 0, 1> since they are clearly independent and span.

Spanning: <-2b - c, b, c> = b<-2, 1, 0> + c <-1, 0, 1>