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If the vector \( a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k} \) and \( \hat{i}+\hat{j}+c \hat{k} \) are colpanar \( (a \neq b \neq c \neq 1) \), then
the value of \( a b c-(a+b+c)= \)
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the value of \( a b c-(a+b+c)= \)
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Verified Answer
The correct answer is:
\( -2 \)
Given vectors, \( a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k} \) and \( \hat{i}+\hat{j}+c \hat{k} \)
For vectors to be coplanar
\[
\begin{array}{l}
\left|\begin{array}{lll}
a & 1 & 1 \\
1 & b & 1 \\
1 & 1 & c
\end{array}\right|=0 \\
\Rightarrow a(b c-1)-1(c-1)+1(1-b)=0 \\
\Rightarrow a b c-a-c+2+1-b=0 \\
\Rightarrow a b c-(a+b+c)=-2
\end{array}
\]
For vectors to be coplanar
\[
\begin{array}{l}
\left|\begin{array}{lll}
a & 1 & 1 \\
1 & b & 1 \\
1 & 1 & c
\end{array}\right|=0 \\
\Rightarrow a(b c-1)-1(c-1)+1(1-b)=0 \\
\Rightarrow a b c-a-c+2+1-b=0 \\
\Rightarrow a b c-(a+b+c)=-2
\end{array}
\]
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