Find normal and tangential components of vector $E_{1}=4\hat{x}+\hat{y}-3\hat{z}$

Question

The surface of separation between regions $1$ and $2$ is a plane whose equation is

$2x+y+z=1$. If $E_{1}=4\hat{x}+\hat{y}-3\hat{z}$, find the normal and tangential

component of $E_{1}$.

Answer 1

The normal vector of the plane is $\frac1{\sqrt6}$(2,1,1). Then, the normal component of $E_1$ is

$$\frac1{\sqrt6}(2,1,1)\cdot(4,1,-3)=\sqrt6$$

and its tangential component is

$$\sqrt{|E_1|^2-(\sqrt6)^2}=\sqrt{26-6}=2\sqrt5$$