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Applications of Cross Product

If a, b and c...

Question

If $\to a, \to b$ and $\to c$ be three non-zero and coplanar vectors, then which of the following statement(s) is/are true?

\to a \times (\to b \times \to c), \to b \times (\to c \times \to a), \to c \times (\to a \times \to b

form a right handed system

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\to c, (\to a \times \to b) \times \to c, \to a \times \to b

form a right handed system

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\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a < 0

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

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\frac{(\to a \times \to b) \cdot (\to b \times \to c)}{(\to b \times \to c) \cdot (\to a \times \to c)} = - 1

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

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Solution

The correct option is D $\frac{(\to a \times \to b) \cdot (\to b \times \to c)}{(\to b \times \to c) \cdot (\to a \times \to c)} = - 1$ if $\to a + \to b + \to c = 0$
For the vectors to form right handed system,
$1)$ they should be non-coplanar
$2)$ they should be perpendicular to each other

For option $(a)$
$\to a \times (\to b \times \to c) = (\to a \cdot \to c) \to b - (\to a \cdot \to b) \to c$ $(i)$
$\to b \times (\to c \times \to a) = (\to b \cdot \to a) \to c - (\to b \cdot \to c) \to a$ $(i i)$
$\to c \times (\to a \times \to b) = (\to c \cdot \to b) \to a - (\to c \cdot \to a) \to b$ $(i i i)$

Adding $(1), (2)$ and $(3)$ , we get sum as zero hence
$\to a \times (\to b \times \to c) = - \to b (\to c \times \to a) - \to c (\to a \times \to b)$
They are coplanar so can't form right handed system

For option $(b)$
$(\to a \times \to b) \times \to c ⊥ \to c$
and $(\to a \times \to b) \times \to c ⊥ \to a \times \to b$
Also they will be non coplanar
So, option $(b)$ forms right handed system

for option $(c)$
$\to a + \to b + \to c = 0$
$\Rightarrow | \to a + \to b + \to c |^{2} = 0$
$| \to a |^{2} + \to a \cdot \to b + \to a \cdot \to c + \to b \cdot \to a$ $+ | \to b |^{2} + \to b \cdot \to c + \to c \cdot \to a + \to c \cdot \to b$ $+ | \to c |^{2} = 0$

$\Rightarrow | \to a |^{2} + \to b |^{2} + \to c |^{2} + 2 (\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a) = 0$
Since, $| \to a |^{2} + \to b |^{2} + \to c |^{2} > 0$
$(\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a) < 0$

For option $(d)$
$\to a + \to b + \to c = 0$
$(\to a + \to b + \to c) \times \to b = 0$
$\Rightarrow \to a \times \to b = \to b \times \to c$ $(i)$
Also, $(\to a + \to b + \to c) \times \to c = 0$
$\Rightarrow - \to a \times \to c = \to b \times \to c$ $(i i)$
putting values of $(i)$ and $(i i)$ in LHS, we get
$\frac{(\to a \times \to b) \cdot (\to b \times \to c)}{(\to b \times \to c) \cdot (\to a \times \to c)} = - 1$
option $(d)$ is correct.

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