How to write an equation for plane, which includes dots with radius-vectors $\mathbf r_{1}, \mathbf r_{2}, \mathbf r_{3}$ that do not lie on a straight line? The answer is
$$ (\mathbf r, ([\mathbf r_{3} , \mathbf r_{1}] + [\mathbf r_{2} , \mathbf r_{3}] + [\mathbf r_{1} , \mathbf r_{2}]) ) = (\mathbf r_{1}, [\mathbf r_{2}, \mathbf r_{3}]). $$
How to explain it?
Thanks to the comments, I think I understand the problem: show that the given formula is an equation for the plane containing the three non-collinear points $r_1,r_2,r_3$. Given a general point $r$ in the plane, $r-r_1$ will be orthogonal to the cross product of $r_2-r_1$ and $r_3-r_1$. That is, $$(r-r_1)\cdot((r_2-r_1)\times(r_3-r_1))=0\tag1$$ which is to say $$r\cdot((r_2-r_1)\times(r_3-r_1))=r_1\cdot((r_2-r_1)\times(r_3-r_1))\tag2$$ Now $v\times v=0$ for all $v$, so $$(r_2-r_1)\times(r_3-r_1)=r_2\times r_3-r_2\times r_1-r_1\times r_3\tag3$$ Also, $u\times v=-v\times u$, and $u\cdot(u\times v)=0$, so (2) becomes $$r\cdot(r_2\times r_3+r_1\times r_2+r_3\times r_1)=r_1\cdot(r_2\times r_3)\tag4$$ and we're done.