Jannek93

2022-10-08

Is there any relation between Gram-Schmidt process in ${\mathbb{R}}^{3}$ 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?

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?

tona6v

Beginner2022-10-09Added 6 answers

Note that the cross-product of two vectors is defined only on ${\mathbb{R}}^{3}$. So, I will assume that we are working on ${\mathbb{R}}^{3}$

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

$\begin{array}{}\text{(1)}& {w}_{3}=\frac{{v}_{1}\times {v}_{2}}{\Vert {v}_{1}\times {v}_{2}\Vert}(={w}_{1}\times {w}_{2}).\end{array}$

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 ${w}_{3}$

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

$\begin{array}{}\text{(1)}& {w}_{3}=\frac{{v}_{1}\times {v}_{2}}{\Vert {v}_{1}\times {v}_{2}\Vert}(={w}_{1}\times {w}_{2}).\end{array}$

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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