Question

If $a+b+c=\alpha dandb+c+d=\beta a$ and a,b,c are non-coplanar vectors , then show that $a+b+c+d=0$

Answer 1

$a+b+c=\alpha d$ [ Given ]

$\Rightarrow$ $b+c=\alpha d-a$ ------ ( 1 )

$b+c+d=\beta a$ [ Given ]

$\Rightarrow$ $b+c=\beta a-d$ ------ ( 2 )

$\Rightarrow$ $\alpha d-a=\beta a-d$ [ From ( 1 ) and ( 2 ) ]

$\Rightarrow$ $\alpha d+d=\beta a+a$

$\Rightarrow$ $(\alpha +1)d=(\beta +1)a$ ----- ( 3 )

$\overrightarrow{d}\parallel \overrightarrow{a}$

Now, $\alpha =-1$ and $\beta =-1$

$\therefore$ Equation ( 3 ) will be $0.$

$\therefore$ $a+b+c+d=0$