Order of Differentiation

Question

I have a UNIT vector n, which I write as $n=[ n_{x},n_{y} ,n_{z} ]$ and a position vector r where $r =[x,y,z]$. When calculating $(\vec{r} \cdot \vec{\bigtriangledown} ) \vec{n}$ I do the following - where the $\vec{\bigtriangledown}$ is the grad operator -

$(x \frac{\partial }{\partial x} +y\frac{\partial }{\partial y} +z\frac{\partial }{\partial z} )[n_{x},n_{y} ,n_{z} ]$

My question is, is this true: $(x \frac{\partial }{\partial x} +y\frac{\partial }{\partial y} +z\frac{\partial }{\partial z} )[n_{x},n_{y} ,n_{z} ] \stackrel{?}{=} ( \frac{\partial x}{\partial x} +\frac{\partial y}{\partial y} +\frac{\partial z}{\partial z} )[n_{x},n_{y} ,n_{z} ]$

If it is not, then why? Thanks

Answer 1

I believe I have just found the answer in Wikipedia https://en.wikipedia.org/wiki/Del under the Precautions section - no wonder why.

$( \vec{r} \cdot \vec{\bigtriangledown} )f \neq ( \vec{\bigtriangledown} \cdot \vec{r} )f$

"Del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function."