If hat{a}, hat{b}, hat{c} are unit vectors,th | vedclass

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(B) We know that for any unit vector $\hat{u}$,$|\hat{u}|^2 = 1$.Expanding the terms,we have $|\hat{a}+\hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 + 2(\hat

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$3$

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(B) We know that for any unit vector $\hat{u}$,$|\hat{u}|^2 = 1$.Expanding the terms,we have $|\hat{a}+\hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 + 2(\hat{a} \cdot \hat{b}) = 1 + 1 + 2(\hat{a} \cdot \hat{b}) = 2 + 2(\hat{a} \cdot \hat{b})$.Similarly,$|\hat{b}+\hat{c}|^2 = 2 + 2(\hat{b} \cdot \hat{c})$ and $|\hat{c}+\hat{a}|^2 = 2 + 2(\hat{c} \cdot \hat{a})$.Summing these,we get $S = |\hat{a}+\hat{b}|^2 + |\hat{b}+\hat{c}|^2 + |\hat{c}+\hat{a}|^2 = 6 + 2(\hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{c} + \hat{c} \cdot \hat{a})$.Consider the identity $|\hat{a} + \hat{b} + \hat{c}|^2 = |\hat{a}|^2 + |\hat{b}|^2 + |\hat{c}|^2 + 2(\hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{c} + \hat{c} \cdot \hat{a}) = 3 + 2(\hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{c} + \hat{c} \cdot \hat{a})$.Since $|\hat{a} + \hat{b} + \hat{c}|^2 \geq 0$,we have $3 + 2(\hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{c} + \hat{c} \cdot \hat{a}) \geq 0$,which implies $2(\hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{c} + \hat{c} \cdot \hat{a}) \geq -3$.Substituting this into the expression for $S$,we get $S \geq 6 - 3 = 3$.Thus,the least value is $3$.

If $\hat{a}, \hat{b}, \hat{c}$ are unit vectors,then the least value of $|\hat{a}+\hat{b}|^2+|\hat{b}+\hat{c}|^2+|\hat{c}+\hat{a}|^2$ will be-

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