(overrightarrow{a}+2 overrightarrow{b}-overri | vedclass

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(C) ધારો કે પદાવલિ $E = (\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{a}-\overrightarrow{b}) 	imes (\overright

Vedclass · Accepted Answer

$3[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$

Vedclass · Answer

(C) ધારો કે પદાવલિ $E = (\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{a}-\overrightarrow{b}) 	imes (\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c}))$ છે.પ્રથમ,ક્રોસ પ્રોડક્ટ પદને સરળ બનાવો: $(\overrightarrow{a}-\overrightarrow{b}) 	imes (\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c})$.$= (\overrightarrow{a} 	imes \overrightarrow{a}) - (\overrightarrow{a} 	imes \overrightarrow{b}) - (\overrightarrow{a} 	imes \overrightarrow{c}) - (\overrightarrow{b} 	imes \overrightarrow{a}) + (\overrightarrow{b} 	imes \overrightarrow{b}) + (\overrightarrow{b} 	imes \overrightarrow{c})$.કારણ કે $\overrightarrow{a} 	imes \overrightarrow{a} = \overrightarrow{0}$ અને $\overrightarrow{b} 	imes \overrightarrow{b} = \overrightarrow{0}$,અને $\overrightarrow{b} 	imes \overrightarrow{a} = -(\overrightarrow{a} 	imes \overrightarrow{b})$ નો ઉપયોગ કરતા:$= \overrightarrow{0} - (\overrightarrow{a} 	imes \overrightarrow{b}) - (\overrightarrow{a} 	imes \overrightarrow{c}) + (\overrightarrow{a} 	imes \overrightarrow{b}) + \overrightarrow{0} + (\overrightarrow{b} 	imes \overrightarrow{c}) = (\overrightarrow{b} 	imes \overrightarrow{c}) - (\overrightarrow{a} 	imes \overrightarrow{c}) = (\overrightarrow{b} 	imes \overrightarrow{c}) + (\overrightarrow{c} 	imes \overrightarrow{a})$.હવે,$E = (\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{b} 	imes \overrightarrow{c}) + (\overrightarrow{c} 	imes \overrightarrow{a}))$.સ્કેલર ટ્રિપલ પ્રોડક્ટની વ્યાખ્યા $[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}] = \overrightarrow{u} \cdot (\overrightarrow{v} 	imes \overrightarrow{w})$ નો ઉપયોગ કરતા:$E = \overrightarrow{a} \cdot (\overrightarrow{b} 	imes \overrightarrow{c}) + \overrightarrow{a} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) + 2\overrightarrow{b} \cdot (\overrightarrow{b} 	imes \overrightarrow{c}) + 2\overrightarrow{b} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) - \overrightarrow{c} \cdot (\overrightarrow{b} 	imes \overrightarrow{c}) - \overrightarrow{c} \cdot (\overrightarrow{c} 	imes \overrightarrow{a})$.નોંધો કે પુનરાવર્તિત સદિશો ધરાવતું કોઈપણ સ્કેલર ટ્રિપલ પ્રોડક્ટ $0$ થાય છે (દા.ત.,$\overrightarrow{a} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) = 0$).$E = [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] + 0 + 0 + 2[\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}] - 0 - 0$.કારણ કે $[\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}] = [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$,તેથી $E = [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] + 2[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] = 3[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$.

$(\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{a}-\overrightarrow{b}) \times (\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c}))$ ની કિંમત શોધો.

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