If \(\hat{a}, \hat{b}, \hat{c}\), are unit vectors and \(\hat{a}+\hat{b}+\hat{c}=0\) then the value of of \(\hat{a}\cdot \hat{b}+\hat{b}\cdot \hat{c}+\hat{c}\cdot \hat{a}\) is :

Concept:

Dot Product: it is also called the inner product or scalar product

Let the two vectors are \(\rm \vec a\) and \(\rm \vec b\)

Dot Product of two vectors is given by:  \(\rm \vec a.{\rm{\;}}\vec b\) = |a||b| cos θ

Where |\(\rm \vec a\)| = Magnitudes of vectors a, |\(\rm \vec b\)| = Magnitudes of vectors b and θ is the angle between a and b

Formulas of Dot Product:

 \(\rm \vec i.\vec i = \vec j.\vec j = \vec k.\vec k = 1\)

\(\rm \vec i.\vec j = \vec j.\vec i = \vec i.\vec k = \vec k.\vec i =\vec j.\vec k= \vec k.\vec j = 0\)

Calculation:

Given that,

(â + b̂ + ĉ) = 0    ----(1)

We know that,

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

⇒ (â + b̂ + ĉ)² = â ⋅ â + b̂ ⋅ b̂ + ĉ ⋅ ĉ + 2(â ⋅ b̂ + b̂ ⋅ ĉ + ĉ ⋅ â) 

From equation (1), we get

⇒ (1 + 1 + 1) + 2 (â ⋅ b̂ + b̂ ⋅ ĉ + ĉ ⋅ â) = 0

∴ (â ⋅ b̂ + b̂ ⋅ ĉ + ĉ ⋅ â) = - 3/2