[a+2 b-c a-b a-b-c]=

MCQ

$[\bar{a}+2 \bar{b}-\bar{c} \quad \bar{a}-\bar{b} \quad \bar{a}-\bar{b}-\bar{c}]=$

A
$\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$
✓
$3\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$
C
$0$
D
$2\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$

✓

Answer

Correct option: B.

$3\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$

(B) $[\overline{ a }+2 \overline{b}-\overline{ c } \quad \overline{ a }-\overline{ b } \quad \overline{ a }-\overline{ b }-\overline{ c }]$
$=(\overline{ a }+2 \overline{b}-\overline{ c }) \cdot\{(\overline{ a }-\overline{ b }) \times(\overline{ a }-\overline{ b }-\overline{ c })\}$
$=(\overline{ a }+2 \overline{b}-\overline{ c })$ $\cdot\{\overline{ a } \times \overline{ a }-\overline{ a } \times \overline{ b }-\overline{ a } \times \overline{ c }-\overline{ b } \times \overline{ a }+\overline{ b } \times \overline{ b }+\overline{ b } \times \overline{ c }\}$
$=(\overline{ a }+2 \overline{b}-\overline{ c })\{\overline{ b } \times \overline{ a }-\overline{ a } \times \overline{ c }-\overline{ b } \times \overline{ a }+\overline{ b } \times \overline{ c }\}$
$=(\overline{ a }+2 \overline{b}-\overline{ c })\{-\overline{ a } \times \overline{ c }+\overline{ b } \times \overline{ c }\}$
$\begin{array}{l}=\left[\begin{array}{lll}\overline{ a } & \overline{ b } & \overline{ c }\end{array}\right]-2\left[\begin{array}{lll}\overline{ b } & \overline{ a } & \overline{ c }\end{array}\right] \\ =\left[\begin{array}{lll}\overline{ a } & \overline{ b } & \overline{ c }\end{array}\right]+2\left[\begin{array}{lll}\overline{ a } & \overline{ b } & \overline{ c }\end{array}\right] \\ =3\left[\begin{array}{lll}\overline{ a } & \overline{ b } & \overline{ c }\end{array}\right]\end{array}$