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If a $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{b}}+\hat{\mathrm{k}}$, and $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{c} \hat{\mathrm{k}}$ are coplanar vectors,
then what is the value of $a+b+c-a b c ?$
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then what is the value of $a+b+c-a b c ?$
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The correct answer is:
2
Vectors a $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{c} \hat{\mathrm{k}}$ are
coplanar vectors. $\Rightarrow\left|\begin{array}{lll}a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c\end{array}\right|=0$
$\Rightarrow$ a $(b c-1)-1(c-1)+1(1-b)=0$
$\Rightarrow \quad a b c-a-c+1+1-b=0$
$\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}-\mathrm{abc}=2$
coplanar vectors. $\Rightarrow\left|\begin{array}{lll}a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c\end{array}\right|=0$
$\Rightarrow$ a $(b c-1)-1(c-1)+1(1-b)=0$
$\Rightarrow \quad a b c-a-c+1+1-b=0$
$\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}-\mathrm{abc}=2$
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