Is there any relation between Gram-Schmidt process in RR^3 and vector cross product? Using Gram-Schmidt orthogonalization process we can find an orthogonal set of vectors from a given set of vectors,also we were taught previously that crossing between two non-collinear vectors gives a vector perpendicular to the two vectors.Is there any correlation between the two processes of find orthogonal system of vectors,are the two essentially the same?

Jannek93 2022-10-08 Answered
Is there any relation between Gram-Schmidt process in R 3 and vector cross product?
Using Gram-Schmidt orthogonalization process we can find an orthogonal set of vectors from a given set of vectors,also we were taught previously that crossing between two non-collinear vectors gives a vector perpendicular to the two vectors.Is there any correlation between the two processes of find orthogonal system of vectors,are the two essentially the same?
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tona6v
Answered 2022-10-09 Author has 6 answers
Note that the cross-product of two vectors is defined only on R 3 . So, I will assume that we are working on R 3
If you have 3 linearly independent vectors v 1 , v 2 and v 3 , if you apply the Gram-Schmidt orthogonalization process to them and you obtain w 1 , w 2 , w 3 , then
(1) w 3 = v 1 × v 2 v 1 × v 2 ( = w 1 × w 2 ) .
So, if you are aware of the cross-product, it is enough to compute w 1 and w 2 and then to simply use (1) to get w 3
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