I'm trying to prove or disprove that if $\vec n(x,y,z)$ is a unit vector, then $(\vec n\cdot\nabla)\vec n$ is orthogonal to $\vec n$. For this I first tried to compute $\vec n\cdot((\vec n\cdot\nabla)\vec n)$ and show that it's zero. I wrote it out in components, but the resulting expression contained only additions, no subtractions at all, so I couldn't simply cancel out something.
Then I wrote $\vec n$ as
$$\vec n=\vec n_0/|\vec n_0|$$
and tried to do the same with its components. But the expression appeared too large to handle manually. I then used Wolfram Mathematica to check that $\vec n\cdot((\vec n\cdot\nabla)\vec n)=0$ symbolically, and it confirmed it.
But I still wonder, are there any ways of proving this without needing to handle large intermediate expressions?