If $[(\overline{a}+2\overline{b}+3\overline{c})\times(\overline{b}+2\overline{c}+3\overline{a})]\cdot(\overline{c}+2\overline{a}+3\overline{b})=54$ then the value of $[\overline{a}\overline{b}\overline{c}]$ is
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Logic Tip: The determinant property is a massive shortcut! Expanding the cross and dot products manually involves multiplying out $3 \times 3 \times 3 = 27$ terms and canceling most of them. Recognizing the coefficient matrix transforms a 5-minute algebraic slog into a 30-second arithmetic calculation.