Question

Let a, b, and c be non-coplanar vectors. If $[a+2b2b+c5c+a]=\lambda \left[abc\right]$, then find $\lambda$

Answer 1

$\left[\begin{array}{ccc}1& 2& 0\\ 0& 2& 1\\ 1& 0& 5\end{array}\right][\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}]=\lambda [\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}]$

$\Rightarrow \left[\begin{array}{ccc}1& 2& 0\\ 0& 2& 1\\ 1& 0& 5\end{array}\right]-\lambda =0$ where $[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}]\ne 0$

$\Rightarrow \lambda =\left|\begin{array}{ccc}1& 2& 0\\ 0& 2& 1\\ 1& 0& 5\end{array}\right|$

$=1(10-0)-2(0-1)+0$

$=10+2=12$

$\therefore \lambda =12$