Let V = 2 i + j - k , W = i + 3 k , | U | = 2 . If U is a vector … - Vidyadip

MCQ

Let $\vec V = 2\hat i + \hat j - \hat k$ , $\vec W = \hat i + 3\hat k$ , $\left| {\vec U} \right| = 2$ . If $\vec U$ is a vector in $x-y$ plane, then greatest value of ${\left( {\left[ {\vec U\,\vec V\,\vec W} \right]} \right)^2}$ is

✓
$232$
B
$340$
C
$236$
D
$312$

✓

Answer

Correct option: A.

$232$

a
Let $|\vec U| = 2\cos \alpha \widehat {\rm{i}} + 2\sin \alpha \widehat {\rm{j}}$

$([\vec{U} \vec{V} \vec{W}])^{2}=\left|\begin{array}{ccc}{2 \cos \alpha} & {2 \sin \alpha} & {0} \\ {2} & {1} & {-1} \\ {1} & {0} & {3}\end{array}\right|^{2}$

$=|6 \cos \alpha-14 \sin \alpha|^{2}$

Maximum value $=36+196=232$

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