Question

Find the point of intersection of the three planes $r\cdot a=1,r\cdot b=1,r\cdot c=1$, where a, b, c are three non coplanar vector.

Answer 1

A point $r$ on the intersection of $3$ planes satisfies $r.\left(a-b\right)=0\&r.\left(a-c\right)=0$

$\therefore$ $r$ is perpendicular to both $(a-b)$ and $(a-c)$

$\therefore$ $r=$ $R\left(a-b\right)\times \left(a-c\right)$

$or,r=R\left(a\times a-a\times c-b\times a+b\times c\right)$

$=R\left(a\times b+b\times c+c\times a\right)$

$Nowr.a=1$

$or,R\left(a\times b+b\times c+c\times a\right)a=1$

$or,R\left(\left(a\times b\right).a+\left(b\times c\right).a+\left(c\times a\right).a\right)=1$

$But\left(a\times b\right).a=\left(c\times a\right).a=0$

$\therefore R=\frac{1}{a.\left(b\times c\right)}=\left(c\times a\right).a=0$

$\therefore R=\frac{1}{a.\left(b\times c\right)}$

$\therefore r=\frac{\left(a\times b+b\times c+c\times a\right)}{a.\left(b\times c\right)}$ is the point of intersection$.$

We get, $r=\frac{\left(a\times b+b\times c+c\times a\right)}{a.\left(b\times c\right)}.$