Anybody know how I can build a orthogonal base using only a vector? I have a vector in the form $v_1 = [a, b, -a, -b]$, where $a$ and $b$ are real numbers. I did try build in the "adhoc way" but, nothing, I only got two orthogonal vectors:
$$v_1 = [a, b, -a, -b], \text{ } v_2 = [a, -b, a, -b]$$
I need more two vectors to complete the orthogonal basis $\{v_1, v_2, v_3, v_4\}$. Anybody can help me?
Thanks…
Best Answer
No need for choosing a basis a priori - you just need one starting vector. There is a straight-forward algorithm that achieves exactly what you asked for:
Pick a vector. WLOG, you chose $(x_1,x_2,x_3,x_4)$. Now write it as a quaternion: $$x_1+ix_2+jx_3+kx_4$$ Then, since multiplication by $i,j,k$ rotates this vector $90^0$ across the various axes of our 4D space, the following three vectors make your initial choice of vector into an orthonormal basis: $$i(x_1+ix_2+jx_3+kx_4)=ix_1-x_2+kx_3-jx_4\mapsto (-x_2,x_1,-x_4,x_3)$$ $$j(x_1+ix_2+jx_3+kx_4)=jx_1-kx_2-x_3+ix_4\mapsto (-x_3,x_4,x_1,-x_2)$$ $$k(x_1+ix_2+jx_3+kx_4)=kx_1+jx_2-ix_3-x_4\mapsto (-x_4,-x_3,x_2,x_1)$$