Question

The value of $\overrightarrow{a}\cdot \left(\right(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\left)\right)$ is equal to

• A. $0$
• B. $2\left[abc\right]$
• C. $\left[abc\right]$
• D. none of these

Answer 1

The correct option is A $0$

$\overrightarrow{a}\cdot \left(\right(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\left)\right)=a.((b\times c)\times a+(b\times c)\times b+(b\times c)\times c)=a.((a.b)c-(a.c)b)=a.((a.b)c-(a.c)b)=(a.b)(a.c)-(a.c)(a.b)=0$