Question

$\overline{a},\overline{b},\overline{c}$ are non-coplanar then $\frac{[\overline{a}+2\overline{b}\overline{b}+2\overline{c}\overline{c}+2\overline{a}]}{\left[\overline{a}\overline{b}\overline{c}\right]}$ is __________.

Answer 1

$[a+2bb+2cc+2a]$

$=(a+2b)\left\{\right(b+2c)\times (c+2a\left)\right\}$

$=(a+2b)\{b\times c+2b\times a+2c\times c+4c\times a\}$

$=a(b\times c)+2a\left(\overrightarrow{b\times a}\right)+4a(\overrightarrow{c}\times a)+2b\left(\overrightarrow{b\times c}\right)+4b\left(\overrightarrow{b\times a}\right)+8b(c\times a)$

$=\left[abc\right]+8\left[bca\right]$

$=\left[abc\right]+8\left[abc\right]$

$=9\left[abc\right]$

$\frac{[a+2bb+2cc+2a]}{\left[abc\right]}=\frac{9\left[abc\right]}{\left[abc\right]}$