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Unit Vectors

If the planes...

Question

If the planes $x - c y - b z = 0$ , $c x - y + a z = 0$ and $b x + a y - z = 0$ pass through a straight line, then find the value of $a^{2} + b^{2} + c^{2} + 2 a b c$ .

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Solution

It they passes through a straight line
$∣ ∣ ∣ ∣ \begin{matrix} 1 & - c & - b c & - 1 & a b & a & - 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 1 (1 - a^{2}) + c (- c - a b) - b (a c + b) = 0 1 - a^{2} - c^{2} - a b c - a b c - b^{2} = 0 a^{2} + b^{2} + c^{2} + 2 a b c = 1$

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