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) }.

s2vunov 2022-09-05 Answered
Consider the vectors u = 2 i + j + k and v = i +2 j .
a) Determine a positive orthornomal basis { a , b , c } with a parallel to u and b coplanar with u and v .
b) Determine the coordinates of w = 3 i +4 j +5 k in the orthonormal basis { a , b , c }.
I'm stuck in how i would find b and c .
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Answers (2)

emarisidie6
Answered 2022-09-06 Author has 7 answers
For b to be coplanar with u,v then it is in the span of u,v. Y
You replaced u with u ~ = u / u . You can just take v, make it orthogonal to u ~ , then normalize.
By "make orthogonal to", I mean take v ~ = v v , u ~ v in the span of u and v so that v ~ , u = 0. This is just Gram-Schmidt.
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tun1ju2k1ki
Answered 2022-09-07 Author has 2 answers
Since you’re working in R 3 , you can use cross products to generate a basis with the requisite properties: u × v is orthogonal to both, while ( u × v ) × u is orthogonal to u and u × v , i.e., it lies in the plane spanned by u and v . Normalize and order these vectors so that the basis has the desired orientation.
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