Which of the following is(are) correct (where \hat{a} , \hat{b} , \hat{c} are unit vectors)?

Engineering

Mathematics

Plane and Its Different Forms

Question

Which of the following is(are) correct (where $\hat{a}, \hat{b}, \hat{c}$ are unit vectors)?

${\vec{v}}_{1} = \hat{i} - 2 \hat{j} + \hat{k}, {\vec{v}}_{2} = 3 \hat{i} + 2 \hat{j} - \hat{k}, {\vec{v}}_{3} = \hat{i} + 2 \hat{j} + \hat{k}, then {\vec{v}}_{1}, {\vec{v}}_{2} and {\vec{v}}_{3} are complanar$

$| \hat{a} + \hat{b} | < 1 then angle between \vec{a} and \vec{b} will be obtuse$

The plane YOZ bisect the line segment joining A(2, 1, 2) and B(–2, 2, 1), (where O is origin)

$| \hat{a} - \hat{b} |^{2} + | \hat{b} - \hat{c} |^{2} + | \hat{c} - \hat{a} |^{2} = 9 then \hat{a} \times \hat{b} = \hat{b} \times \hat{c} = \hat{c} \times \hat{a}$

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(A) $| \hat{a} - \hat{b} |^{2} + | \hat{b} - \hat{c} |^{2} + | \hat{c} - \hat{a} |^{2}$

$= maximum = 9 when = \hat{a}^{\land} \hat{b} = \frac{2 π}{3}$

$= \hat{b}^{\land} \hat{c} = \hat{c}^{\land} \hat{a} .$

(B) 1 + 1 + 2cos q < 1, $- 1 \leq cosθ < \frac{- 1}{2}$

$\Rightarrow θ \in (\frac{2 π}{3}, π] .$

(D) $| \begin{matrix} 1 & - 2 & 1 \\ 3 & 2 & - 1 \\ 1 & 2 & 1 \end{matrix} | = 2 + 6 + 2 - (2 - 2 - 6) = 16 .$