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Multiplication with Vectors

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Question

If $\to a$ , $\to b$ , $\to c$ are three vectors and $\to a + \to b + \to c = 0$ prove that $\to a \times \to b = \to b \times \to c = \to c \times \to a$

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Solution

$\begin{matrix} \to a + \to b + \to c = 0 \to a \times (\to a + \to b + \to c) = \to a \times 0 \to a \times \to a + \to a \times \to b + \to a \times \to c = 0 \to a \times \to b + \to a \times \to c = 0 \to a \times \to b - \to c \times \to a = 0 (\to a \times \to a = 0 & \to a \times \to b = - \to b \times \to a) \to a \times \to b = \to c \times \to a - - - - - (1) \end{matrix}$

$\begin{matrix} \to b \times (\to a + \to b + \to c) = \to b \times 0 \to b \times \to a + \to b \times \to b + \to b \times \to c = 0 - \to a \times \to b + \to b \times \to c = 0 (\to a \times \to a = 0 & \to a \times \to b = - \to b \times \to a) \to a \times \to b = \to b \times \to c - - - - - (2) \end{matrix}$

From $(1)$ and $(2)$

$\to a \times \to b = \to b \times \to c = \to c \times \to a$

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Similar questions

\to a, \to b, \to c

are three vectors such that

\to a \times \to b = \to c, \to b \times \to c = \to a, \to c \times \to a = \to b

then prove that

| \to a | = | \to b | = | \to c |

\to a, \to b, \to c

are three vectors such that

\to a, \to b, \to c

, then prove that

\to a \times \to b = \to b \times \to c = \to c \times \to a

, and hence show that

[\to a \to b \to c] = 0.

\to a, \to b, \to c

are three vectors such that

\to a + \to b + \to c = \to 0,

then prove that

\to a \times \to b = \to b \times \to c = \to c \times \to a,

and hence show that

[\to a \to b \to c] = 0.

\to A

\to B

\to C

are vectors such that

| \to B | =

\to C |

[(\to A + \to B) \times (\to A + \to C)] \times (\to B \times \to C) . (\to B + \to C) = 0

\to a + \to b + \to c = 0

, the prove that:

\to a \times \to b = \to b \times \to c = \to c \times \to a

\to a, \to b, \to c

are non-zero vectors.

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