Question

Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes Let $\overrightarrow{ a }=3 \hat{ i }+\hat{ j }-\hat{ k },$ $\overrightarrow{ b }=\hat{ i }+ b _2 \hat{ j }+ b _3 \hat{ k }, b _2, b _3 \in R ,$ $\overrightarrow{ c }= c _1 \hat{ i }+ c _2 \hat{ j }+ c _3 \hat{ k }, c _1, c _2, c _3 \in R$ be three vectors such that $b_2 b_3>0, \vec{a} \cdot \vec{b}=0$ and \begin{pmatrix}0 & -c_3 & c_2 \\\c_3 & 0 & -c_1 \\\\-c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix} 1 \\\b_2 \\\b_3\end{pmatrix}=\begin{pmatrix}3-c_1 \\\1-c_2 \\\\-1-c_3\end{pmatrix} . Then, which of the following is/are TRUE?

• A. $\vec{a}.\vec{c}=0$
• B. $\vec{b}.\vec{c}=0$
• C. $\left | \vec{b} \right |> \sqrt10$
• D. $\left | \vec{c} \right |\leq \sqrt10$

Answer 1

Answer: (B), (C), (D)

$\vec{b}.\vec{c}=0$

Answer 2

The correct answer is option
(B) $\vec{b}.\vec{c}=0$
(C) $\left | \vec{b} \right |> \sqrt10$
(D): $\left | \vec{c} \right |\leq \sqrt10$