Discussion Forum

If 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 ≠ 1) are coplanar, then find $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$

Answer:

Explanation:

Since vectors are coplanar

.'. $\begin{vmatrix}a & 1 & 1 \\1 & b &1\\1&1&c \end{vmatrix}$ = 0

$\begin{vmatrix}a & 1 & 1 \\1-a & b-1 &0\\0&1-b&c -1\end{vmatrix}$ =0 [Using R₂ - R_1, R₃ - R₂]

→ a(b - 1)(c- 1)-(1 -a) {(c- 1)-(1 -b)} = 0

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

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

→  (1 -b) (1 -c) + (1 - a) (1 - c) + (1 - a) (1 - b) = (1 - a)(1 - b)(1 - c)

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