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STD 12 - 10. vector algebra — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 10. vector algebra4 Marks
Question
If $\left[ {a \times b\;\;b \times c\;\;c \times a} \right] = \alpha \;{\left[ {a\;\;b\;\;c} \right]^2}$ then $\lambda$ is equal to 
✓

Answer

b
L. H.S.

$=(\bar{a} \times \bar{b}) \cdot[(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})]$

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

$ = (\vec a \times \vec b) \cdot [[\bar b\bar c\bar a]\vec c]\quad [\vec b \times \vec c.\vec c = 0]$

$=[\bar{a} \bar{b} \bar{c}] \cdot(\bar{a} \times \bar{b} \cdot \bar{c})=[\bar{a} \bar{b} \bar{c}]^{2}$

$[\vec{a} \times \vec{b} \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]=[\bar{a} \bar{b} \bar{c}]^{2}$

So $\lambda=1$

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