Answer:
Given; \[\vec{R}=\left( \vec{A}+\vec{B} \right)\] Taking dot product of \[\overset{\to }{\mathop{R}}\,\]with
itself we have,
\[\overset{\to
}{\mathop{R}}\,.\overset{\to }{\mathop{R}}\,=\left( \overset{\to
}{\mathop{A}}\,+\overset{\to }{\mathop{B}}\, \right).\left( \overset{\to
}{\mathop{A}}\,+\overset{\to }{\mathop{B}}\, \right)\]\[{{R}^{2}}=\overset{\to
}{\mathop{A}}\,.\overset{\to }{\mathop{A}}\,+2\overset{\to }{\mathop{A}}\,.\overset{\to
}{\mathop{B}}\,+\overset{\to }{\mathop{B}}\,.\overset{\to
}{\mathop{B}}\,\,\,\text{or}\]\[{{R}^{2}}=\overset{\to
}{\mathop{A}}\,+2\overset{\to }{\mathop{A}}\,\overset{\to }{\mathop{B}}\,\cos
\theta +{{B}^{2}}\]
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