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Question
For any two vectors \[\vec{a} \text{ and } \vec{b}\] write when \[\left| \vec{a} + \vec{b} \right| = \left| \vec{a} \right| + \left| \vec{b} \right|\] holds.
Solution
\[\text{ Given that }\]
\[\left| \vec{a} + \vec{b} \right| = \left| \vec{a} \right| + \left| \vec{b} \right|\]
\[\text{ Squaring both sides,we get }\]
\[ \left| \vec{a} + \vec{b} \right|^2 = \left( \left| \vec{a} \right| + \left| \vec{b} \right| \right)^2 \]
\[ \Rightarrow \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + 2 \vec{a} . \vec{b} = \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + 2\left| \vec{a} \right| \left| \vec{b} \right|\]
\[ \Rightarrow \vec{a} . \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right|\]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = \left| \vec{a} \right| \left| \vec{b} \right|.............. (\text{ where } \theta \text{ is the angle between } \vec{a} \text{ and } \vec{b} )\]
\[ \Rightarrow \cos \theta = 1\]
\[ \Rightarrow \theta = 0^o \]
\[ \Rightarrow \vec{a} \text{ and } \vec{b} \text{ are parallel }.\]
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