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 \( \left(a_{1}, a_{2}, a_{3}\right),\left(b_{1}, b_{2}, b_{3}\right) \) and \( \left(c_{1}, c_{2}, c_{3}\right) \) are linearly independent and
\[
x\left(a_{1}, a_{2}, a_{3}\right)+y\left(b_{1}, b_{2}, b_{3}\right)+z\left(c_{1}, c_{2}, c_{3}\right)=0
\]
then
(a) \( x=y=z \)
(b) \( x=y=z=0 \)
(c) \( x+y+z=0 \)
(d) \( x+y+z \neq 0 \)
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