Is there any relation between Gram-Schmidt process in $\mathbb R^3$ and vector cross product?

Question

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?

Answer 1

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$$w_3=\frac{v_1\times v_2}{\lVert v_1\times v_2\rVert}(=w_1\times w_2).\tag1$$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 $w_3$.

Answer 2

The answer is more no than yes.

In Gram-Schmidt 3D, the first step is to construct a vector $v_2$ orthogonal to $u_1$, using a linear combination of $u_1$ and $u_2$. This vector is coplanar with them. It has little to do with $u_1\times u_2$.

The second step is to construct a vector $v_3$ orthogonal to both $u_1$ and $u_2$, using a linear combination of $u_1,u_2,u_3$. This vector is indeed orthogonal to $v_1$ and $v_2$ but this is unavoidable. And it doesn't recover the length of $v_1\times v_2$.