# Consider the vectors vec(u) = 2 vec(i) +vec(j) +vec(k) and vec(v) = vec(i) +2 vec(j) . a) Determine a positive orthornomal basis {vec(a) , vec(b) , vec(c) } with vec(a) parallel to vec(u) and vec(b) coplanar with vec(u) and vec(v) . b) Determine the coordinates of vec(w) = 3 vec(i) +4 vec(j) +5 vec(k) in the orthonormal basis { vec(a) , vec(b) , vec(c) }.

Consider the vectors $\stackrel{\to }{u}$ = 2$\stackrel{\to }{i}$ +$\stackrel{\to }{j}$ +$\stackrel{\to }{k}$ and $\stackrel{\to }{v}$ = $\stackrel{\to }{i}$ +2 $\stackrel{\to }{j}$.
a) Determine a positive orthornomal basis {$\stackrel{\to }{a}$ ,$\stackrel{\to }{b}$ ,$\stackrel{\to }{c}$ } with $\stackrel{\to }{a}$ parallel to $\stackrel{\to }{u}$ and $\stackrel{\to }{b}$ coplanar with $\stackrel{\to }{u}$ and $\stackrel{\to }{v}$ .
b) Determine the coordinates of $\stackrel{\to }{w}$ = 3$\stackrel{\to }{i}$ +4 $\stackrel{\to }{j}$ +5 $\stackrel{\to }{k}$ in the orthonormal basis {$\stackrel{\to }{a}$ , $\stackrel{\to }{b}$ , $\stackrel{\to }{c}$ }.
I'm stuck in how i would find $\stackrel{\to }{b}$ and $\stackrel{\to }{c}$.
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emarisidie6
For b to be coplanar with u,v then it is in the span of u,v. Y
You replaced u with $\stackrel{~}{u}=u/‖u‖$. You can just take v, make it orthogonal to $\stackrel{~}{u}$, then normalize.
By "make orthogonal to", I mean take $\stackrel{~}{v}=v-⟨v,\stackrel{~}{u}⟩v$ in the span of u and v so that $⟨\stackrel{~}{v},u⟩=0$. This is just Gram-Schmidt.
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tun1ju2k1ki
Since you’re working in ${\mathbb{R}}^{3}$, you can use cross products to generate a basis with the requisite properties: $\stackrel{\to }{u}×\stackrel{\to }{v}$ is orthogonal to both, while $\left(\stackrel{\to }{u}×\stackrel{\to }{v}\right)×\stackrel{\to }{u}$ is orthogonal to $\stackrel{\to }{u}$ and $\stackrel{\to }{u}×\stackrel{\to }{v}$ , i.e., it lies in the plane spanned by $\stackrel{\to }{u}$ and $\stackrel{\to }{v}$ . Normalize and order these vectors so that the basis has the desired orientation.