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Question
Let \[\vec{u,} \vec{v} \text{ and } \vec{w}\] be vectors such \[\vec{u} + \vec{v} + \vec{w} = \vec{0} .\] If \[\left| \vec{u} \right| = 3, \left| \vec{v} \right| = 4 \text{ and } \left| \vec{w} \right| = 5,\] then find \[\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .\]
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Solution
\[\text{ Given that }\]
\[ \vec{u} + \vec{v} + \vec{w} = 0\]
\[ \Rightarrow \left| \vec{u} + \vec{v} + \vec{w} \right| = 0\]
\[ \Rightarrow \left| \vec{u} + \vec{v} + \vec{w} \right|^2 = 0\]
\[ \Rightarrow \left| \vec{u} \right|^2 + \left| \vec{v} \right|^2 + \left| \vec{w} \right|^2 + 2\left( \vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u} \right) = 0\]
\[ \Rightarrow 3^2 + 4^2 + 5^2 + 2\left( \vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u} \right) = 0 .................(\text{ Given }: \left| \vec{u} \right| = 3, \left| \vec{v} \right| = 4 \text{ and } \left| \vec{w} \right| = 5)\]
\[ \Rightarrow 9 + 16 + 25 + 2\left( \vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u} \right) = 0\]
\[ \Rightarrow 50 + 2\left( \vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u} \right) = 0\]
\[ \Rightarrow 2\left( \vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u} \right) = - 50\]
\[ \therefore \vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u} = \frac{- 50}{2} = - 25\]
