If \(\vec{a}+\vec{b}+\vec{c} = \vec{0}\), \(|\vec{a}| = 3\), \(|\vec{b}| = 4\) and \(|\vec{c}| = 5\), then the value of \(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → −25
Since \(\vec{a}+\vec{b}+\vec{c} = \vec{0}\), its scalar product with itself is zero:
\((\vec{a}+\vec{b}+\vec{c})\cdot(\vec{a}+\vec{b}+\vec{c}) = 0\)
\(|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}) = 0\)
\(9+16+25+2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}) = 0\)
\(2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}) = -50\)
\(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a} = -25\)