A set of vector \( \left\{\left(a_{1}, a_{2}, a_{3}\right),\left(b_{1}, b_{2}, b_{3}\right),\left(c_{1}, c_{2}, c_{3}\right)\right\} \) is said to be linearly independent if and only if
\[
\left|\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right| \neq 0,
\]
otherwise the set is said to be linearly dependent.
A similar result holds for \( \left\{\left(a_{1}, a_{2}\right),\left(b_{1}, b_{2}\right)\right\} \).
If \( \left(1, a, a^{2}\right),\left(1, b, b^{2}\right) \) and \( \left(1, c, c^{2}\right) \) are linearly independent, then
(a) \( a+b+c \neq 0 \)
(b) \( (b-a)(c-b) \neq 0 \)
(c) \( (b-c)(c-a)(a-b) \neq 0 \)
(d) none of these
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