If the vectors ai^+j^+k^,i^+bj^+k^,i^+j^+c^k^(a≠1,b≠1,c≠1) are coplanar, then the value of11−a+11−b+11−c is

a 1 11 b 11 1 c=0⇒a111−ab−101−a0c−1=0 [Applying R2→R2−R1 and R3→R3−R1 ] Dividing C1 by 1−a,C2 by 1−b and C3 by 1−c , we get a1−a11−b−1c−11−1010−1=0⇒a1−a+11−b+11−c=0Adding 1 to the first term, we get11−a+11−b+11−c=1.

If the vectors ai^+j^+k^,i^+bj^+k^,i^+j^+c^k^(a≠1,b≠1,c≠1) are coplanar, then the value of11−a+11−b+11−c is

a 1 11 b 11 1 c=0⇒a111−ab−101−a0c−1=0 [Applying R2→R2−R1 and R3→R3−R1 ] Dividing C1 by 1−a,C2 by 1−b and C3 by 1−c , we get a1−a11−b−1c−11−1010−1=0⇒a1−a+11−b+11−c=0Adding 1 to the first term, we get11−a+11−b+11−c=1.

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

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Detailed Solution

$|\begin{array}{lll} a 1 1 \\ 1 b 1 \\ 1 1 c \end{array}| = 0$

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

$[Applying R_{2} \to (R_{2} - R_{1}) and R_{3} \to (R_{3} - R_{1})]$

Dividing $C_{1} by 1 - a, C_{2} by 1 - b and C_{3} by 1 - c$ , we get

$|\begin{array}{ccc} \frac{a}{1 - a} & \frac{1}{1 - b} & - \frac{1}{c - 1} \\ 1 & - 1 & 0 \\ 1 & 0 & - 1 \end{array}| = 0 \Rightarrow \frac{a}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = 0$