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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors2 Marks
MCQ
If $\bar{a}+\bar{b}+\bar{c}=\overline{0}$, then which relation is correct?
  • A
    $\overline{ a }=\overline{ b }=\overline{ c }=0$
  • B
    $\overline{ a } \cdot \overline{ b }=\overline{ b } \cdot \overline{ c }=\overline{ c } \cdot \overline{ a }$
  • ✓
    $\overline{ a } \times \overline{ b }=\overline{ b } \times \overline{ c }=\overline{ c } \times \overline{ a }$
  • D
    None of these
✓

Answer

Correct option: C.
$\overline{ a } \times \overline{ b }=\overline{ b } \times \overline{ c }=\overline{ c } \times \overline{ a }$
(C) Since $\bar{a}+\bar{b}+\bar{c}=\overline{0}$
$\begin{array}{l}\Rightarrow \overline{ a } \times(\overline{ a }+\overline{ b }+\overline{ c })=0 \\ \Rightarrow \overline{ a } \times \overline{ a }+\overline{ a } \times \overline{ b }+\overline{ a } \times \overline{ c }=0\end{array}$
$\Rightarrow \overline{ a } \times \overline{ b }=-\overline{ a } \times \overline{ c }=\overline{ c } \times \overline{ a }$ ...(i)
Similarly, $\overline{ b } \times(\overline{ a }+\overline{ b }+\overline{ c })=0$
$\Rightarrow \overline{ a } \times \overline{ b }=\overline{ b } \times \overline{ c }$ ...(ii)
By (i) and (ii), we get
$\overline{a} \times \overline{b}=\overline{b} \times \overline{c}=\overline{c} \times \overline{a}$

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