0
$\begingroup$

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?

  • 0
    I do not know what you mean by the notation $(x,y)$, nor by the notation $[x,y]$. Without knowing what these notations mean, it will be difficult to explain why that equation gives a plane.2012-12-04
  • 0
    $(\mathbf x , \cdot \mathbf y)$ means the dot product of vectors x, y. $[\mathbf x , \cdot \mathbf y ]$ means the cross product of these vectors.2012-12-04
  • 0
    Thanks. You will know that one form of the equation of a plane is $ax+by+cz=d$, which can be written $r\cdot s=t\cdot u$, where $r=(x,y,z)$, $s=(a,b,c)$, and $t,u$ are any two vectors chosen so that $t\cdot u=d$. I don't see how to relate $s,t,u$ to $r_1,r_2,r_3$, but maybe I don't understand whether $r_1,r_2,r_3$ are arbitrary or instead have some special properties in connection with this plane.2012-12-04
  • 0
    Maybe, there was used a complanarity of vectors, which are a combination of $\mathbf r_{1}, r_{2}, r_{3}$. So, that's like $$((\mathbf r - \mathbf r_{1}), [(\mathbf r_{2} - \mathbf r_{1}), (\mathbf r_{3} - \mathbf r_{1})]) = 0,$$ if I'm right.2012-12-04

1 Answers 1

1

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.