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This can be a little subtle the first time you see it, so I'll move through the rationale carefully: $\langle (\delta \mathbf{r} \cdot \nabla )^2 \rangle _{vac} = \langle (\delta x \ \partial_x + \delta y\ \partial_y + \delta {z}\ \partial_z )^2\rangle _{vac}$ On expanding you get squared and cross terms. The text mentions $\langle \delta \mathbf r ... 2 The scalar product is just a shortcut notation for multiplication and then addition: $$\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z\tag{1}$$ It's commutative if the underlying multiplication is commutative, and otherwise it is not. Notation like$\vec{a}\cdot\nabla\$ is not really a scalar product, but it takes the same form of (1) and applies it to ...