Question

If the vectors $\overrightarrow{p}=a\hat{i}+\hat{j}+\hat{k},\overrightarrow{q}=\hat{i}+b\hat{j}+\hat{k},\overrightarrow{r}=\hat{i}+\hat{j}+c\hat{k}$ are coplanar, then for $a,b,c\ne 1$ show that
$\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$

Answer 1

Given, $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ coplanar

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

$c_1+c_1-c_2,c_2+c_2-c_3$

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

=$-(1-a)(1-b)(1-c)+(1-a)(1-b)+(1-a)(1-c)+(1-b)(1-c)=0$

Dividing $(1-a)(1-b)(1-c)$

= $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}-1=0$

= $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$