Find the expression of the tetrahedron passing through these points: $P_0=(0,0,0),P_1=(a,0,0),P_2=(0,b,0),P_3=(0,0,c) \qquad a,b,c>0$

Question

Finding the plane passing through $P_1,P_2,P_3$:

$$\det \begin{pmatrix} x-a & y & z \\ -a & b & 0 \\ -a & 0 & c \end{pmatrix}=-a(-b \ z)+c ((x-a) \ b+ a \ y)=cb \ x+ac \ y +ab \ z-abc $$

So, the expression is: $$T=\{ cb \ x+ac \ y +ab \ z \le abc, x \ge 0, y \ge 0, z \ge 0 \}$$

Is it correct?

Thanks!