Question

A vector equation of the line of intersection of the planes $r=b+{\lambda }_1(b-a)+{\mu }_1(a+c)$
$r=c+{\lambda }_2(b-c)+{\mu }_1(a+b)$ $a,b,c$ being non-coplanar vectors is.

• A. $r=a+{\mu }_1(b+c)$
• B. $r=b+{\mu }_1(a+2c)$
• C. $r=a+{\mu }_1(b+2c)$
• D. $r=b+{\mu }_1(a+c)$

Answer 1

Answer: (D)

$r=b+{\mu }_1(a+c)$

At points of intersection of the two planes, we have
$b+{\lambda }_1(b-a)+{\mu }_1(a+c)=c+{\lambda }_2(b-c)+{\mu }_2(a+b)$
$\Rightarrow (-{\lambda }_1+{\mu }_1-{\mu }_2)a+(1+{\lambda }_1-{\lambda }_2-{\mu }_2)b+({\mu }_1-1+{\lambda }_2)c=0$
As $a,b,c$ are non-coplanar, we have
$-{\lambda }_1+{\mu }_1-{\mu }_2=0,1+{\lambda }_1-{\lambda }_2-{\mu }_2)=0,{\mu }_1-1+{\lambda }_2=0$
Eliminating ${\lambda }_2,{\mu }_2$ i.e., writing ${\lambda }_2=1-{\mu }_1$ from the last equation in the second equation, we have

${\lambda }_1=0,{\mu }_2={\mu }_1$ So the required line is $r=b+{\mu }_1(a+c)$