Completing System of Vectors to an Orthogonal Basis I want to complete ((1, -2, 2, -3), (2, -3, 2, 4)) to an orthgonal basis.  I honestly don't really know how to do this.  This a practice problem to a test, were we haven't done problems related to this.  I know that an orthogonal basis has that for basis $(v_1, ... , v_n)$, $v_i\perp v_j$ for $i\ne j$.  So I'm assuming I'm working in $R^4$, as the question only specifies "Euclideam Spcae".  In terms of actually solving the problem, I'm strugging, thanks in advance!

here is what you can do. you already have two vectors $u = (1, -2, 2, -3)^T, v = (2,- 2, 3, 4)^T$ that are orthogonal. we will first find four orthogonal vectors; making them of unit length is easier.
what we will do is pick a vector $a$ and find the projection on to the space spanned by $u, v$ and subtract it. we can just start with $a = (1,0,0,0).$  so let $$a = ku + lv + w $$ with $l,k$ are to be chosen so that $u^Tw = v^Tw  = 0.$  that is $$k = u^Ta/u^Tu, l = v^Ta/v^Tv \to k = \frac 1{14}, l = \frac 1{33},\\ w =(1,0,0,0)^T - \frac 1{14}((1, -2, 2, -3)^T + \frac1{33}(2,- 2, 3, 4)^T$$ 
once you have $w,$ or a nice multiple of it,  we have $u, v, w$ orthogonal. now you can use another vector hopefully $(0,1,0,0)$ and do the same trick to find the fourth vector. 
