If the vectors $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} + a\widehat{\dot{\mathbf{j}}} + a^{2}\widehat{\mathbf{k}},\ \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + b\widehat{\dot{\mathbf{j}}} + b^{2}\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}} = \widehat{\dot{\mathbf{i}}} + c\widehat{\dot{\mathbf{j}}} + c^{2}\widehat{\mathbf{k}}$ are three non-coplanar vectors and $\left| \begin{matrix} a & a^{2} & 1 + a^{3} \ b & b^{2} & 1 + b^{3} \ c & c^{2} & 1 + c^{3} \end{matrix} ight| = 0,\ then\ the\ value\ of\ abc\ is$

Question

If the vectors $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} + a\widehat{\dot{\mathbf{j}}} + a^{2}\widehat{\mathbf{k}},\ \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + b\widehat{\dot{\mathbf{j}}} + b^{2}\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}} = \widehat{\dot{\mathbf{i}}} + c\widehat{\dot{\mathbf{j}}} + c^{2}\widehat{\mathbf{k}}$ are three non-coplanar vectors and $\left| \begin{matrix} a & a^{2} & 1 + a^{3} \ b & b^{2} & 1 + b^{3} \ c & c^{2} & 1 + c^{3} \end{matrix} ight| = 0,\ then\ the\ value\ of\ abc\ is$

Dr. Abubacker · Accepted Answer

(d)
Given vectors are non-coplanar, if
$\mathrm{\Delta} = \left| \begin{matrix}
1 & a & a^{2} \
1 & b & b^{2} \
1 & c & c^{2}
\end{matrix} ight| 
eq 0$
$Now,\left| \begin{matrix}
a & a^{2} & 1 + a^{3} \
b & b^{2} & 1 + b^{3} \
c & c^{2} & 1 + c^{3}
\end{matrix} ight| = 0$
$\Longrightarrow \left| \begin{matrix}
a & a^{2} & 1 \
b & b^{2} & 1 \
c & c^{2} & 1
\end{matrix} ight| + \left| \begin{matrix}
a & a^{2} & a^{3} \
b & b^{2} & b^{3} \
c & c^{2} & c^{3}
\end{matrix} ight| = 0\$
$\Longrightarrow \ \mathrm{\Delta}(1 + abc) = 0\  \Longrightarrow abc = - 1$

Explanation

Reviewed by Dr. Abubacker

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