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If the vectors $\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+a \hat{\mathbf{j}}+a^{2} \hat{\mathbf{k}}$,
$\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+b \hat{\mathbf{j}}+b^{2} \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+c \hat{\mathbf{j}}+c^{2} \hat{\mathbf{k}}$ are three
non-coplanar vectors and $\left|\begin{array}{lll}a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0$,
then the value of $a b c$ is
Options:
$\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+b \hat{\mathbf{j}}+b^{2} \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+c \hat{\mathbf{j}}+c^{2} \hat{\mathbf{k}}$ are three
non-coplanar vectors and $\left|\begin{array}{lll}a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0$,
then the value of $a b c$ is
Solution:
2743 Upvotes
Verified Answer
The correct answer is:
$-1$
Since, $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$, and $\overrightarrow{\mathbf{c}}$ are non-coplanar vectors,
therefore $[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}] \neq 0$
$\Rightarrow \Delta=\left|\begin{array}{lll}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{array}\right| \neq 0$
$\Rightarrow$
$\Delta \neq 0$
$$
\begin{array}{l}
\text { Now, } & \left(\begin{array}{lll}
a & a^{2} & 1+a^{3} \\
b & b^{2} & 1+b^{3} \\
c & c^{2} & 1+c^{3}
\end{array} \mid=0\right. \\
\Rightarrow\left|\begin{array}{lll}
a & a^{2} & 1 \\
b & b^{2} & 1 \\
c & c^{2} & 1
\end{array}\right|+\left|\begin{array}{lll}
a & a^{2} & a^{3} \\
b & b^{2} & b^{3} \\
c & c^{2} & c^{3}
\end{array}\right|=0 \\
\Rightarrow & \quad \Delta(1+a b c)=0
\end{array}
$$
$\Rightarrow$
$$
a b c=-1
$$
therefore $[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}] \neq 0$
$\Rightarrow \Delta=\left|\begin{array}{lll}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{array}\right| \neq 0$
$\Rightarrow$
$\Delta \neq 0$
$$
\begin{array}{l}
\text { Now, } & \left(\begin{array}{lll}
a & a^{2} & 1+a^{3} \\
b & b^{2} & 1+b^{3} \\
c & c^{2} & 1+c^{3}
\end{array} \mid=0\right. \\
\Rightarrow\left|\begin{array}{lll}
a & a^{2} & 1 \\
b & b^{2} & 1 \\
c & c^{2} & 1
\end{array}\right|+\left|\begin{array}{lll}
a & a^{2} & a^{3} \\
b & b^{2} & b^{3} \\
c & c^{2} & c^{3}
\end{array}\right|=0 \\
\Rightarrow & \quad \Delta(1+a b c)=0
\end{array}
$$
$\Rightarrow$
$$
a b c=-1
$$
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