The deformation gradient $\mathbf F$ describes how material line elements change their length and orientation during deformation; the velocity gradient $\mathbf L$ describes the rate of these changes.
We can write $\mathbf L=\mathbf D+\mathbf W$, where $\mathbf D=(\mathbf L+\mathbf L^T)/2$ (thus symmetric) and $\mathbf W=(\mathbf L-\mathbf L^T)/2$ (thus skewsymmetric).
In words, $\mathbf D$ is the Eulerian strain-rate tensor and $\mathbf W$ is the body spin.
Now suppose $\mathbf n$ is a unit vector field.
We have $\mathbf n\cdot(\mathbf L^T\mathbf n)=\mathbf n\cdot(\mathbf D^T\mathbf n)+\mathbf n\cdot(\mathbf W^T\mathbf n)=\mathbf n\cdot[(\mathbf D^T+\mathbf W^T)\mathbf n]=\mathbf n\cdot(\mathbf D\mathbf n)$.
But I do not understand why the last equation is true.
I would appreciate any help or hint. Thank you.