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Condition for Coplanarity of Four Points

If a, b,c a...

Question

If $¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c$ are non-coplanar vectors, then show that the vectors $- ¯ ¯ ¯ a + 3 ¯ ¯ b - 5 ¯ ¯ c, - ¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c$ and $2 ¯ ¯ ¯ a - 3 ¯ ¯ b + ¯ ¯ c$ are coplanar

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Solution

$\to a, \to b, \to c$ are non-coplanar vectors
$∣ ∣ ∣ ∣ \begin{matrix} - 1 & 3 & - 5 - 1 & 1 & 1 2 & - 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ \begin{matrix} = - 2 (1 + 3) - 3 (- 1 - 2) - 5 (3 - 2) = - 4 + 9 - = 0 \end{matrix}$
They are coplanar.

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