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Multiplication with Vectors

If a⃗+b⃗+c⃗...

Question

If $\to a + \to b + \to c = \to 0$ show that $\to a \times \to b = \to b \times \to c = \to c \times \to a$ .

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Solution

Given that,
$\begin{matrix} \to a + \to b + \to c = \to 0 \Rightarrow \to a \times (\to a + \to b + \to c) = \to 0 \times \to a \Rightarrow \to a \times \to a + \to a \times \to b + \to a \times \to c = \to 0 \to a \times \to a = 0 \Rightarrow 0 + \to a \times \to b + \to a \times \to c = \to 0 \Rightarrow \to a \times \to b = - \to a \times \to c \Rightarrow \to a \times \to b = \to c \times \to a . . . . . . . . . (i i) [- \to a \times \to c = \to c \times \to a] \end{matrix}$
Similarly,
$\begin{matrix} \to a + \to b + \to c = \to 0 \Rightarrow \to b \times (\to a + \to b + \to c) = \to b \times \to 0 \Rightarrow \to b \times \to a + \to b \times \to b + \to b \times \to c = \to 0 [\to b \times \to b = 0] \Rightarrow 0 + \to b \times \to a + \to b \times \to c = \to 0 \Rightarrow \to b \times \to c = - \to b \times \to a \Rightarrow \to b \times \to c = \to a \times \to b . . . . . . . . . (i i) [- \to b \times \to a = \to a \times \to b] \end{matrix}$
Now, from $(i)$ & $(i i)$ we have,
$\to a \times \to b = \to b \times \to c = \to c \times \to a$

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