If the vectors \(a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}\) , \((a,b,c e1)\) are coplanar, then \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\)
Coplanar means \(\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} = 0\).
Do \(R_2 \to R_2 - R_1\) and \(R_3 \to R_3 - R_1\). Write \(p = 1 - a\), \(q = 1 - b\), \(r = 1 - c\), so \(b - 1 = -q\) and \(c - 1 = -r\). Expanding:
\(a(-q)(-r) - 1(-p)(-r)\cdots\). Simplifying gives \(qr + rp + pq = pqr\), using \(a = 1 - p\).
Divide by \(pqr\): \(\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1\), that is, \(\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} = 1\).
Answer: (B) 1
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