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Vector Algebra question

2022 · Shift 2 · Q31

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PYQ Library
JEE Advanced
Mathematics
Vector Algebra
2022 · Shift 2 · Q31

Vector Algebra question

2022 · Shift 2 · Q31

JEE AdvancedMathematicsVector AlgebraMultiple correct+4 / −2

Let

\hat{\imath}, \hat{\jmath}

and

\hat{k}

be the unit vectors along the three positive coordinate axes. Let

\begin{aligned} & \vec{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k} \text {, } \\ & \vec{b}=\hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R} \text {, } \\ & \vec{c}=c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}, \quad c_{1}, c_{2}, c_{3} \in \mathbb{R} \end{aligned}

be three vectors such that

b_{2} b_{3}\gt 0, \vec{a} \cdot \vec{b}=0

and

\left(\begin{array}{ccc} 0 & -c_{3} & c_{2} \\ c_{3} & 0 & -c_{1} \\ -c_{2} & c_{1} & 0 \end{array}\right)\left(\begin{array}{l} 1 \\ b_{2} \\ b_{3} \end{array}\right)=\left(\begin{array}{r} 3-c_{1} \\ 1-c_{2} \\ -1-c_{3} \end{array}\right) .

Then, which of the following is/are TRUE?

A
$\vec{a} \cdot \vec{c}=0$
B
$\vec{b} \cdot \vec{c}=0$
C
$|\vec{b}|\gt \sqrt{10}$
D
$|\vec{c}| \leq \sqrt{11}$

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