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) }.



Answered question


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 .

Answer & Explanation



Beginner2022-09-06Added 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.


Beginner2022-09-07Added 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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?