Let $T : R3 → R3$ be a linear transformation which projects vectors onto the plane π with equation $x−y + 2z = 0$. Find a matrix $A$ such that $T = T_A$

I know how to complete the question I just had a quick inquiry. So I know I need to choose two vectors that span the plane and are orthogonal. It is easy to choose vectors that span the plan but is there a quick and easy way to find two vectors that span the plane and are orthogonal without just guessing what possible combination would work? I know that the two vectors in this case are 1 1 0 and 1 -1 -1

  • $\begingroup$ @Arthur how so? $\endgroup$ – mp12345 Nov 14 '16 at 22:06
  • $\begingroup$ I know to pick x and y but just figuring out if they are orthogonal might take a bit longer if I was in a time crunch on an exam $\endgroup$ – mp12345 Nov 14 '16 at 22:07
  • $\begingroup$ Sorry, I read your question wrong. $\endgroup$ – Arthur Nov 14 '16 at 22:08

To get two orthogonal vectors in the plane, first find one vector in the plane (say $(1,1,0)$), then take the cross product with the normal vector $(1,-1,2)$: $$ (1,1,0)\times(1,-1,2)=(1\cdot2-(-1)\cdot0,0\cdot1-1\cdot2, 1\cdot(-1)-1\cdot1)\\ =(2,-2,-2) $$ which may be scaled back to $(1,-1,-1)$.

  • $\begingroup$ if I was to just leave it without scaling back is that fine? $\endgroup$ – mp12345 Nov 14 '16 at 22:18
  • $\begingroup$ Of course. Simplifying is just personal preference. $\endgroup$ – Arthur Nov 14 '16 at 22:45
  • $\begingroup$ perfect thanks again $\endgroup$ – mp12345 Nov 14 '16 at 22:53

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