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Scalar Or Dot Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSScalar Or Dot Product2 Marks
Question
For any two vectors $\vec{\text{a}} $ and $\vec{\text{b}},$ write when $\big|\vec{\text{a}}+\vec{\text{b}}\big|=|\vec{\text{a}}|+\big|\vec{\text{b}}\big|$ holds.
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Answer

Given that  $\big|\vec{\text{a}}+\vec{\text{b}}\big|=|\vec{\text{a}}|+\big|\vec{\text{b}}\big|$ Squaring both sides, we get $\big|\vec{\text{a}}+\vec{\text{b}}\big|^2=\big(|\vec{\text{a}}|+\big|\vec{\text{b}}\big|\big)^2$ $\Rightarrow|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2+2\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2+2|\vec{\text{a}}|\big|\vec{\text{b}}\big|$ $\Rightarrow\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}|\big|\vec{\text{b}}\big|$ $\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=|\vec{\text{a}}|\big|\vec{\text{b}}\big|$ (where $\theta$ is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$) $\Rightarrow\cos\theta=1$ $\Rightarrow\theta=0^{\circ}$$\Rightarrow\vec{\text{a}}$ and $\vec{\text{b}}$ are parallel.

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