Given three unit vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\), such that \(\mathbf{a} + \mathbf{b} + \mathbf{c} = 0\), find \(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}\).

a.(a +b + c ) = 0 => a.b + a.c = - a.a = - // a // ^2 = - 1 ( a is a unit vector )=> a.b + c.a = - 1 ; where // // is Euclidean length; ( * )
b.( a + b + c ) = 0 => b.a + b.c = - b.b => a.b + b.c = - // b // ^2 = -1 ( b is a unit vector ); ( * * )
c.( a + b + c ) = 0 => c.a + c.b = -c.c => a.c + b.c = - // c // ^ 2 = - 1 ; ( c is a unit vector ) ( * * * )
From ( * ), ( * * ), ( * * * ) => 2( a. b + b.c + c.a ) = - 3 => a.b + b.c + c.a = - 3 / 2 = - 1.5

Answered by crisforp | 2024-06-10

The value of a ⋅ b + b ⋅ c + c ⋅ a is − 2 3 ​ given the unit vectors and the condition a + b + c = 0 .
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Answered by crisforp | 2024-12-26