Having a plane equation equaled to zero for calculating its angle with another plane I got to calculate the angle between two planes.
$$x-3y+2z=14\\
-x+y+z=10$$
As far as I am concerned, that's simply the angle between their normal vectors. And such vectors are $(1,-3,2)$ and $(-1,1,1)$. Therefore
$$\cos^{-1}\left(\frac{(1,-3,2)\cdot(-1,1,1)}{||(1,-3,2)||\cdot||(-1,1,1)||}\right)$$
Which is like $1.88$ radians.
Now then, to verify this I looked for online calculators, and I couldn't help but notice that all the calculators I found required to input the planes like this:
$$x+y+z=\color{red}{0}$$
See, they ask for plane equations to be equaled to zero. My questions are, then:


*

*Do I need my plane equation to be equaled to zero to calculate such angle?

*Is there an advantage/disadvantage to having an equation equaled to zero?

*Is there a way to simplify my equation to equal it to zero?

 A: Setting the equation to zero doesn't affect the computation of the angle, because, as you said, it only depends on the normal vectors, which are given by the coefficients. 
This means that for the websites you can just input $$ x-3y+2z=0\\-x+y+z=0,$$
and you will get the same angle, because these planes are parallel to yours.
However, note that you can always write the equations in the form $$a(x-x_0)+b(y-y_0)+c(z-z_0)=0,$$ where $\langle a,b,c\rangle$ is the normal vector and $(x_0,y_0,z_0)$ is a point on the plane.
For example, let's write $x−3y+2z=14$ in that form:
$x-3y+2z=14 \iff 0= x-3y+2x-14=(x-14)-3(y-0)+2(z-0)$.
This is not the only way to write it (but it was the easiest). Since $-14=3(2)-2(10)$ another possibility is $0=x-3(y-2)+2(z-10)$.
The main advantage of writing the equation in this form is that you can immediately read off not only the coordinates of the normal vector, but also points (in the example, $(14,0,0)$ and $(0,2,10)$) that are in the plane.
A: You can equate those equations of yours to zero by removing the constants at right hand side. This does not alter the angle because this is just a linear parallel shift. Angles won't change. Homogenous equation (planes passing through origin) are easier to deal with. The constant on the right side just shifts the level surface(plane) up or down the z-axis in 3-D space while keeping it parrallel to the original. Hence while calculating angle those constants do not play any role
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