Let B = {, , }. Applying the Gram-Schmidt algorithm gives the following orthogon

Question

Let B = {, , }. Applying the Gram-Schmidt algorithm gives the following orthogonal (not orthonormal) basis derived from B, under the usual dot product: {, , ) Performing no calculations at all (no cross products, systems, etc.), give an equation for the plane spanned by and . Explain how you did it, and why it works. For each of the following sets of 3 vectors, A) Verify that the first two vectors are orthogonal, and then B) use the Gram-Schmidt process to find an orthonormal basis that spans the same space as the original.

Explanation / Answer

it is 4x + 3y +2z = 0

since the third vector <4,3,2> is perpendicular to both first<-1,0,2> and second vectors<6,-10,3>

and since <2,-2,-1> is a linear combination of <-1,0,2> + <6,-10,3> [ look at the procedure of orthogonalisation ----how we got <6,-10,3> ] <2,-2,-1> is also perpendicular to <4,3,2>.

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