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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors2 Marks
MCQ
For any three vectors $\bar{a}, \bar{b}$ and $\bar{c}$, $(\bar{a}-\bar{b}) \cdot[(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})]$ is equal to :
  • A
    $2 \bar{a} \cdot(\bar{b} \times \bar{c})$
  • B
    $[\bar{a} \bar{b} \bar{c}]$
  • C
    $[\overline{ a } \overline{ b } \overline{ c }]^2$
  • ✓
    $0$
✓

Answer

Correct option: D.
$0$
(D) $(\bar{a}-\bar{b}) \cdot[(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})]$
$=(\bar{a}-\bar{b})$ $\cdot[\overline{ b } \times \overline{ c }+\overline{ b } \times \overline{ a }+\overline{ c } \times \overline{ c }+\overline{ c } \times \overline{ a }]$
$=\bar{a} \cdot(\bar{b} \times \bar{c})+\bar{a} \cdot(\bar{b} \times \bar{a})+\bar{a} \cdot(\bar{c} \times \bar{a})$ $-\overline{ b } \cdot(\overline{ b } \times \overline{ c })-\overline{ b } \cdot(\overline{ b } \times \overline{ a })-\overline{ b } \cdot(\overline{ c } \times \overline{ a })$
$=[\overline{ a } \overline{ b } \overline{ c }]-[\overline{ a } \overline{ b } \overline{ c }]=0$

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