If ( a b ) c = a ( b c ) where a, b and c are any three vector su… - Vidyadip

MCQ

If $\left( {\vec a \times \vec b} \right) \times \vec c = \vec a \times \left( {\vec b \times \vec c} \right)$ where $\vec a, \vec b$ and $\vec c$ are any three vector such that $\vec a \cdot \vec b \ne 0,\vec b \cdot \vec c \ne 0$ then $\vec a$ and $\vec c$ are

A
inclined at an angle of $60^\circ $ between them
B
inclined at an angle of $30^\circ $ between them
C
perpendicular
✓
parallel

✓

Answer

Correct option: D.

parallel

d
$(\vec{a} \times \vec{b}) \times \vec{c}$

$=\vec{a} \times(\vec{b} \times \vec{c}), \vec{a} \cdot \vec{b} \neq 0, \vec{b} \cdot \vec{c} \neq 0$

$\Rightarrow(\vec{a} \cdot \vec{c}) \cdot \vec{b}-(\vec{b} \cdot \vec{c}) \vec{a}$

$=(\vec{a} \cdot \vec{c}) \cdot \vec{b}-(\vec{a} \cdot \vec{b}) \cdot \vec{c}$

$\Rightarrow(\vec{a} \cdot \vec{b}) \cdot \vec{c}=(\vec{b} \cdot \vec{c}) \vec{a}$

$\Rightarrow \vec{a} \| \vec{c}$

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