Question
Define dot product of two vectors and give its geometrical interpretation.
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Answer
Dot product of two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ is defined as, $\vec{\text{a}}.\vec{\text{b}}$ $=|\vec{\text{a}}||\vec{\text{b}}|\cos\theta$ where $\theta$ is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$ Let $\vec{\text{OA}}=\vec{\text{a}},\ \vec{\text{OB}}=\vec{\text{b}}$ and $\angle\text{AOB}=\theta$ Then, $\text{OT}=|\vec{\text{b}}|\cos\theta$
$\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}||\vec{\text{b}}|\cos\theta$ $=\text{OA}\times\text{OT}$ $\vec{\text{a}}.\vec{\text{b}}=\text{OA}.$ Projection of $\vec{\text{b}}$ on $\vec{\text{a}}.$
$\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}||\vec{\text{b}}|\cos\theta$ $=\text{OA}\times\text{OT}$ $\vec{\text{a}}.\vec{\text{b}}=\text{OA}.$ Projection of $\vec{\text{b}}$ on $\vec{\text{a}}.$
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