If a = i + j + k, , , b = i - j + k, , , c = i + 2 j + k , then t… - Vidyadip

MCQ

If $\vec a = \hat i + \hat j + \hat k,\,\,\vec b = \hat i - \hat j + \hat k,\,\,\vec c = \hat i + 2\hat j + \hat k$ , then the value of $\left| {\begin{array}{*{20}{c}}
{\vec a.\vec a}&{\vec a.\vec b}&{\vec a.\vec c} \\
{\vec b.\vec a}&{\vec b.\vec b}&{\vec b.\vec c} \\
{\vec c.\vec a}&{\vec c.\vec b}&{\vec c.\vec c}
\end{array}} \right|$ is

A
$2$
B
$4$
✓
$16$
D
$64$

✓

Answer

Correct option: C.

$16$

c
$[\bar{a} \bar{b} \bar{c}]^{2}=\left|\begin{array}{lll}{\bar{a} \cdot \bar{a}} & {\bar{a} \cdot \bar{b}} & {\bar{a} \cdot \bar{c}} \\ {\bar{b} \cdot \bar{a}} & {\bar{b} \cdot \bar{b}} & {\bar{b} \cdot \bar{c}} \\ {\bar{c} \cdot \bar{a}} & {\bar{c} \cdot \bar{b}} & {\bar{c} \cdot \bar{c}}\end{array}\right|$

and $[\bar{a} \bar{b} \bar{c}]=\left|\begin{array}{ccc}{1} & {1} & {1} \\ {1} & {-1} & {1} \\ {1} & {2} & {-1}\end{array}\right|=4$

$\Rightarrow[\bar{a} \bar{b} \bar{c}]^{2}=16$

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