If w = ( a b ) + ( b c ) + ( c a ), [ a, b, c ] = 2 and w. ( a + … - Vidyadip

MCQ

If $\vec w = \alpha \left( {\vec a \times \vec b} \right) + \beta \left( {\vec b \times \vec c} \right) + \gamma \left( {\vec c \times \vec a} \right),$ $\left[ {\vec a,\vec b,\vec c} \right] = 2$ and $\vec w.\left( {\vec a + \vec b + \vec c} \right) = 8$, then $\alpha + \beta + \gamma =$

A
$64$
✓
$4$
C
$32$
D
$8$

✓

Answer

Correct option: B.

$4$

b
$\overrightarrow {\rm{w}} .\overrightarrow {\rm{c}} = \alpha [\overrightarrow {\rm{a}} \overrightarrow {\rm{b}} \overrightarrow {\rm{c}} ]$

$\therefore \alpha = \frac{1}{2}(\overrightarrow {\rm{w}} .\overrightarrow {\rm{c}} )$

Similarly $\beta = \frac{1}{2}(\overrightarrow w \cdot \overrightarrow a ),\gamma = \frac{1}{2}(\overrightarrow w \cdot \overrightarrow b )$

$\therefore \alpha + \beta + \gamma = \frac{1}{2}(\vec w \cdot \vec a + \vec w \cdot \vec b + \vec w \cdot \vec c) = \frac{1}{2}(8) = 4$

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