Question

If the vectors $a\hat{i}+\hat{j}+\hat{k},$ $\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ $(a,b,c\ne 0)$ are coplanar, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

Answer 1

Given : $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}(a,b,c\ne 0)$ are coplanar.
$\Rightarrow \left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=0$
Applying $C_1\to C_1-C_2$ and $C_2\to C_2-C_3$
$\Rightarrow \left|\begin{array}{ccc}a-1& 0& 1\\ 1-b& b-1& 1\\ 0& 1-c& c\end{array}\right|=0$
Now expanding, we get
$0-(1-c)\left(\right(a-1)-(1-b\left)\right)+c(a-1)(b-1)=0$
$\Rightarrow c(a-1)(b-1)+(1-b)(1-c)-(1-c)(a-1)=0$
$\Rightarrow \frac{c}{1-c}+\frac{1}{1-a}+\frac{1}{1-b}=0$
$\Rightarrow \frac{c}{1-c}+1+\frac{1}{1-a}+\frac{1}{1-b}=1$
$\therefore \frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$