Question

If $a,b,c$ are three non-coplanar vectors such that $a+b+c=\alpha d$ and $b+c+d=\beta a$ then $a+b+c+d$ is equal to

• A. $0$
• B. $\alpha a$
• C. $\beta b$
• D. $(\alpha +\beta )c$

Answer 1

The correct option is A $0$

$a+b+c=\alpha (\beta a-b-c)$
$\Rightarrow (1-\alpha \beta )a+(1+\alpha )b+(1+\alpha )c=0$
$\Rightarrow 1=\alpha \beta$ and $\alpha =-1(a,b,c$ are non-coplanar so linearly independent)
Substitute the value $\alpha =-1,$ we get $a+b+c+d=0$