If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}| = 5, |\vec{b}| = 12$ and $|\vec{c}| = 13 and \vec{a} + \vec{b} + \vec{c} = 0$ find the value of $\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} .$

Solution

Considering dot product on both sides,

$(\vec{a} + \vec{b} + \vec{c}) . (\vec{a} + \vec{b} + \vec{c}) = 0 . 0 \Rightarrow {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} + 2 (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = 0 \Rightarrow 5^{2} + 12^{2} + 13^{2} + 2 (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = 0 \Rightarrow 2 (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = - 338 \Rightarrow (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = \frac{- 338}{2} = - 169 .$

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Vector Algebra

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}| = 5, |\vec{b}| = 12$ and $|\vec{c}| = 13 and \vec{a} + \vec{b} + \vec{c} = 0$ find the value of $\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} .$

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Vector Algebra

If a→ , b→ , c→ are three vectors such that a→ = 5, b→ = 12 and c→ = 13 and a→ + b→ + c→ = 0 find the value of a→ . b→ + b→ . c→ + c→ . a→ .

Some More Questions From Vector Algebra Chapter

Mock Test Series

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Entrance Exams Preparation

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}| = 5, |\vec{b}| = 12$ and $|\vec{c}| = 13 and \vec{a} + \vec{b} + \vec{c} = 0$ find the value of $\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} .$