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Condition for Coplanarity of Four Points

Show that the...

Question

Show that the vectors $\to a \cdot \to b$ and $\to c$ are coplanar if $\to a + \to b \cdot \to b + \to c$ and $\to c + \to a$ are coplanar

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Solution

For $\to a + \to b, \to b + \to c, \to c + \to a$ to be coplanar, their scalar triple product must be zero.
Thus, $(\to a + \to b) . [(\to b + \to c) \times (\to c + \to a)] = 0$
$\Rightarrow (\to a + \to b) . [\to b \times \to c + \to b \times \to a + \to c \times \to c + \to c \times \to a] = 0$
$\Rightarrow \to a . (\to b \times \to c) + 0 + 0 + 0 + 0 + 0 + 0 + \to b . (\to c \times \to a) = 0$
$\Rightarrow 2 [\begin{matrix} \to a & \to b & \to c \end{matrix}] = 0$
Hence, proved.

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Condition for Coplanarity of Four Points

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