Under rotation of coordinates, the newly defined operator transforms as: U ( R ) † † ( V → → ⋅ ⋅ W → → ) U ( R ) = U ( R ) † † ( ∑ ∑ i = 1 3 V i ^ ^ W i ^ ^ ) U ( R ) = ∑ ∑ i = 1 3 ( U ( R ) † † V ^ ^ i U ( R ) ) ( U ( R ) † † W ^ ^ i U ( R ) ) = ∑ ∑ i = 1 3 ( ∑ ∑ j = 1 3 R i j V ^ ^ j ⋅ ⋅ ∑ ∑ k = 1 3 R i k W ^ ^ k ) Rearranging terms and using transpose of rotation matrix as its inverse property: U ( R ) † † ( V → → ⋅ ⋅ W → → ) U ( R ) = ∑ ∑ k = 1 3 ∑ ∑ j = 1 3 ( ∑ ∑ i = 1 3 R j i T R i k ) V ^ ^ j W ^ ^ k = ∑ ∑ k = 1 3 ∑ ∑ j = 1 3 δ δ j , k V ^ ^ j W ^ ^ k = ∑ ∑ i = 1 3 V ^ ^ i W ^ ^ i Where the RHS is the V → → ⋅ ⋅ W → → operator originally defined. Since the dot product defined is invariant under rotation transformation, it is said to be a scalar operator.