A plane (formed by a linear equation) contains a line in 3D space. The linear equation has different multiples that produce the same exact plane. Does that then mean that there are an infinite number of planes that contain that line, or would that be the same plane given that it covers identical lines in 3D space?
I initially thought that there were an infinite number of planes containing that line, but I'm not so sure all of a sudden. To preface, I'm doing this to get a better understanding on my college homework.
Sorry if I didn't explain myself well. Thanks for any help given.
I think it would be useful to point out here what is the difference between the linear equation and the plane itself.
Any non-trivial linear equation in three variables with coefficients in $\mathbb{R}$ defines a plane in $\mathbb{R}^3$ (i.e. in $3D$ space); such a plane is nothing but the solution set (in $\mathbb{R}^3$) to this linear equation. Now note that when you scale your linear equation by any non-zero real number, the solution set for this new scaled equation is exactly the same as the previous one, so the scaled equation defines the same plane , i.e. the same solution set.
On the other hand, given any line in $\mathbb{R}^3$ there are always infinitely many (in fact, uncountably many!) distinct planes containing that line. The intuition for this is easily represented in this picture:
One obtains infinitely many planes passing through the green line by rotating any plane containing the line along the axis given by this same line on an angle $\theta$ (for $0 < \theta \leq 2\pi$).