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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA1 Mark
MCQ
Choose the correct answer from the given four options.If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are three vectors such that $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec{0}$ and $|\vec{\text{a}}|=2,|\vec{\text{b}}|=3$ and $|\vec{\text{c}}|=5,$ then the value of $\vec{\text{a}}\cdot\vec{\text{b}}+\vec{\text{b}}\cdot\vec{\text{c}}+\vec{\text{c}}\cdot\vec{\text{a}}$ is:
  • A
    0.
  • B
    1.
  • ✓
    -19.
  • D
    38.
✓

Answer

Correct option: C.
-19.
Here, $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec{0}$ and $\vec{\text{a}}^2=4,\vec{\text{b}}^2=9,\vec{\text{c}}^2=25$

$\therefore(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}})\cdot(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}})=0$

$\Rightarrow\vec{\text{a}}^2+\vec{\text{a}}\cdot\vec{\text{b}}+\vec{\text{a}}\cdot\vec{\text{c}}+\vec{\text{b}}\cdot\vec{\text{a}}+\vec{\text{b}}^2+\vec{\text{b}}\cdot\vec{\text{c}}+\vec{\text{c}}\cdot\vec{\text{a}}+\vec{\text{c}}\cdot\vec{\text{b}}+\vec{\text{c}}^2=0$

$\Rightarrow\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}+2(\vec{\text{a}}\cdot\vec{\text{b}}+\vec{\text{b}}\cdot\vec{\text{c}}+\vec{\text{c}}\cdot\vec{\text{a}})=0$ $[\because\vec{\text{a}}\cdot\vec{\text{b}}=\vec{\text{b}}\cdot\vec{\text{a}}]$

$\Rightarrow4+9+25+2(\vec{\text{a}}\cdot\vec{\text{b}}+\vec{\text{b}}\cdot\vec{\text{c}}+\vec{\text{c}}\cdot\vec{\text{a}})=0$

$\Rightarrow\vec{\text{a}}\cdot\vec{\text{b}}+\vec{\text{b}}\cdot\vec{\text{c}}+\vec{\text{c}}\cdot\vec{\text{a}}=0$

$\Rightarrow\frac{-38}{2}=-19$

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