If $\vec{a}, \vec{b}$ and $\vec{c}$ are three non coplanar vectors, then $(\vec{a}+\vec{b}+\vec{c})

Question

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three non coplanar vectors, then $(\vec{a}+\vec{b}+\vec{c}) \cdot[(\vec{a}+\vec{b}) 	imes(\vec{a}+\vec{c})]$ equals\

Professor PC Thomas Classes · Answer

(d) : $(\vec{a}+\vec{b}+\vec{c}) \cdot[(\vec{a}+\vec{b}) 	imes(\vec{a}+\vec{c})]$\ $=(\vec{a}+\vec{b}+\vec{c}) \cdot[\vec{a} 	imes \vec{a}+\vec{a} 	imes \vec{c}+\vec{b} 	imes \vec{a}+\vec{b} 	imes \vec{c}]$\ $=(\vec{a}+\vec{b}+\vec{c}) \cdot[\vec{a} 	imes \vec{c}+\vec{b} 	imes \vec{a}+\vec{b} 	imes \vec{c}]$\ $[\because \vec{a} 	imes \vec{a}=0]$\ $=\vec{a} \cdot \vec{a} 	imes \vec{c}+\vec{a} \cdot \vec{b} 	imes \vec{a}+\vec{a} \cdot \vec{b} 	imes \vec{c}+\vec{b} \cdot \vec{a} 	imes \vec{c}+\vec{b} \cdot \vec{b} 	imes \vec{a} +\vec{b} \cdot \vec{b} 	imes \vec{c}+\vec{c} \cdot \vec{a} 	imes \vec{c}+\vec{c} \cdot \vec{b} 	imes \vec{a}+\vec{c} \cdot \vec{b} 	imes \vec{c}$\ $=\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}ight]-\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}ight]-\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}ight]=-\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}ight]$\