Step 1:

The given equation of the plane is $\overrightarrow r.(2\hat i+3\hat j-4\hat k)=1$-----(1)

We know that $\overrightarrow r=x\hat i+y\hat j+z\hat k$

Substituting for $\overrightarrow r$ in equ(1) we get,

$(x\hat i+y\hat j+z\hat k).(2\hat i+3\hat j-4\hat k)=1$

Step 2:

We know that $\hat i.\hat i=\hat j.\hat j=\hat k.\hat k=1$

$\Rightarrow 2x+3y-4z=1$

This equation is the required Cartesian equation of the plane.