Can I write a random unit vector with unit vector components?

Question

Let's say we have a xyz system in 3D and there lies a random plane with its normal vector. We can find the unit vector in the direction of the normal vector by dividing with its magnitude. Could we express this unit vector in i (unit vector on x axis), j(unit vector on y axis) and k(unit vector on z axis) components? If yes, what will the coefficients be in each axis? If the unit vector in the direction of the normal vector has magnitude 1, then the coefficients must be less than 1 (pythagorean theorem), right? I'm talking about vector projections on the x, y and z axis of a unit vector that it is in the direction of the normal vector of a surface. Can we do that, can we split a unit vector even further and find its coefficients on the i, j and k main unit vectors? Why am I thinking about it? Let's say we have a tetrahedron with its 3 sides on the xy, yz, xz plane (they all meet at (0,0,0)). The 3 sides are the projection of the fourth one which has a normal vector n and makes an angle θ with the bottom plane. My teacher said that : dot product of n and k unit vectors equals (cosθ)=(the projection of n on the z axis). And then I wondered how can we measure the projection of the unit vector on the z axis. But, I guess that we are talking about the NORMALIZED vector, so the k coefficient is indeed the projection on the z axis.

Answer 1

The plane $2x+3y+6z =24$ has normal vector $(2,3,6)$. Normalizing it to a unit vector yields

$$\hat n =\frac {(2,3,6)}{\|(2,3,6)\|} =(2/7,3/7,6/7)$$

$2/7$ is the coefficient of $\hat n$ along the $\hat i$ main unit vector.