When I go through examples of calculating the intersection of two planes, there seems to be a convention of choosing an arbitrary point in order to solve the linear equations in question and get a particular point on the intersection line.
Let's take an example from a previously-asked question, where the given equations were:
$x + 2y + z – 1 = 0$
$2x + 3y – 2z + 2 = 0$
And the relevant part in the second-most voted answer goes as follows:
Next, we need to find a particular point on the line. We can try $y=0$
and solve the resulting system of linear
equations:$$\begin{align}x+z-1&=&0\\2x-2z+2&=&0\end{align}$$ giving
$x=0, z=1$
My question:
How does one know what is the correct point to choose, and how does one validates that the chosen point is correct?
Also, if the chosen point is wrong, how does one successfully guess the next point?
I've found another answer that seems to be very relevant, but I can't explain it:
Sometimes the line of intersection happens to be parallel to the z=0
plane. In that case you could try y=0 or x=0. (One of these is sure to
work.)
I'm a beginner, so that's a fundamental thing, probably trivial, I don't get: Why is that true?
Best Answer
There is not "the" correct point. The goal is to find any specific point on the line.
The solver has chosen to set $y=0$ because that results in two equations in two unknowns ($x$ and $z$), which is an easy system to solve.
You could just as easily have picked any other value for any of the variables, but the numbers might have been not as easy to work with (that variable would become a constant, and you would have to move it to the other side in each equation, etc.)--so it's easier if you just make a variable vanish by setting it equal to zero.
The reason specifying a particular value (like $0$) for one of the variables usually works is that the "typical" line is not parallel to any of the axes, so it will contain points with any given value (the one you specify) for any one of the coordinates. But if you are unlucky and the line line is parallel to one of the axes, then there may be no point on the line with that particular coordinate value. In that case, you would find that the resulting system of 2 equations in 2 unknowns has no solution; then you would just do it over, picking a specific value for one of the other variables instead.