Evaluate: vec{a} cdot {(vec{b} + vec{c}) time | vedclass

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(A) We need to evaluate the expression $\vec{a} \cdot \{(\vec{b} + \vec{c}) 	imes (\vec{a} + \vec{b} + \vec{c})\}$.Using the distributive property of

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(A) We need to evaluate the expression $\vec{a} \cdot \{(\vec{b} + \vec{c}) 	imes (\vec{a} + \vec{b} + \vec{c})\}$.Using the distributive property of the cross product:$(\vec{b} + \vec{c}) 	imes (\vec{a} + \vec{b} + \vec{c}) = \vec{b} 	imes \vec{a} + \vec{b} 	imes \vec{b} + \vec{b} 	imes \vec{c} + \vec{c} 	imes \vec{a} + \vec{c} 	imes \vec{b} + \vec{c} 	imes \vec{c}$.Since $\vec{x} 	imes \vec{x} = 0$,this simplifies to:$= \vec{b} 	imes \vec{a} + \vec{b} 	imes \vec{c} + \vec{c} 	imes \vec{a} + \vec{c} 	imes \vec{b}$.Now,taking the dot product with $\vec{a}$:$= \vec{a} \cdot (\vec{b} 	imes \vec{a}) + \vec{a} \cdot (\vec{b} 	imes \vec{c}) + \vec{a} \cdot (\vec{c} 	imes \vec{a}) + \vec{a} \cdot (\vec{c} 	imes \vec{b})$.Using the definition of the scalar triple product $[\vec{x} \vec{y} \vec{z}] = \vec{x} \cdot (\vec{y} 	imes \vec{z})$:$= [\vec{a} \vec{b} \vec{a}] + [\vec{a} \vec{b} \vec{c}] + [\vec{a} \vec{c} \vec{a}] + [\vec{a} \vec{c} \vec{b}]$.Since any scalar triple product with two identical vectors is $0$:$= 0 + [\vec{a} \vec{b} \vec{c}] + 0 + [\vec{a} \vec{c} \vec{b}]$.Using the property $[\vec{a} \vec{c} \vec{b}] = -[\vec{a} \vec{b} \vec{c}]$:$= [\vec{a} \vec{b} \vec{c}] - [\vec{a} \vec{b} \vec{c}] = 0$.

Evaluate: $\vec{a} \cdot \{(\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c})\}$

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