If a, b, c are non- coplanar vectors, then [ lll a +2 b & a + c &… - Vidyadip

MCQ

If $\bar{a}, \bar{b}, \bar{c}$ are non- coplanar vectors, then $\left[\begin{array}{lll}\overline{ a }+2 \overline{b} & \overline{ a }+\overline{ c } & \overline{ b }\end{array}\right]=$

A
$0$
B
$[\overline{ a } \overline{ b } \overline{ c }]$
✓
$-[\overline{ a } \overline{ b } \overline{ c }]$
D
$2[\overline{ a } \overline{ b } \overline{ c }]$

✓

Answer

Correct option: C.

$-[\overline{ a } \overline{ b } \overline{ c }]$

(C) $\left[\begin{array}{lll}\overline{ a }+2 \overline{b} & \overline{ a }+\overline{ c } & \overline{ b }\end{array}\right]$
$=\left[\begin{array}{lll}\overline{ a } & \overline{ a }+\overline{ c } & \overline{ b }\end{array}\right]+\left[\begin{array}{lll}2 \overline{b} & \overline{ a }+\overline{ c } & \overline{ b }\end{array}\right]$
$=[\overline{ a } \overline{ a } \overline{ b }]+[\overline{ a } \overline{ c } \overline{ b }]+[2 \overline{b} \overline{ a } \overline{ b }]+[2 \overline{b} \overline{ c } \overline{ b }]$
$\begin{array}{l}=0-[\overline{ a } \overline{ b } \overline{ c }]+2(0)+2(0) \\ =-[\overline{ a } \overline{ b } \overline{ c }]\end{array}$

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