If \( \vec{a},\vec{b},\vec{c} \) are unit vectors such that \( \vec{a}+\vec{b}+\vec{c}=\vec{0} \). The value of \( \vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a} \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( -\displaystyle\frac{3}{2} \)
Since \( \vec{a}+\vec{b}+\vec{c}=\vec{0} \), the square of its magnitude is 0:
\( |\vec{a}+\vec{b}+\vec{c}|^{2}=0 \)
\( |\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a})=0 \)
Each vector is a unit vector, so each square is 1:
\( 3+2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a})=0 \)
\( \displaystyle \vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}=-\frac{3}{2} \)
NCERT Exemplar:
Vector Algebra
