Question

If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}$ $(a\ne b\ne c\ne 1)$ are coplanar, then value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

• A. $1$
• B. $-1$
• C. $0$
• D. None of these

Answer 1

The correct option is A $1$

Since the given vectors are coplanar, we have

$\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=0$

$\Rightarrow a(bc-1)-1(c-1)+1(1-b)=0$

$\Rightarrow abc-a-c+1+1-b=0$

We need $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$
$=\frac{1-b-c+bc+1-a-c+ac+1-a-b+ab}{1-b-c+bc-a+ab+ac-abc}$
$=\frac{ab+bc+ac-1-2abc}{ab+bc+ac-1-2abc}$

$=1$