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If $ax + by + cz + d = 0$ is Plane $1$ and $a'x + b'y + c'z + d' = 0$ is Plane $2$, then what does (Plane $1$) + $\lambda$(Plane $2$) signify?

I got this doubt when the equation of a line was given as an intersection of two planes Plane $1$ and Plane $2$. And the general equation of a plane passing through that line was described to be of the form "(Plane $1$) + $\lambda$(Plane $2$)".

How do we arrive at this conclusion? My problem is that I am not able to imagine $\lambda$ times a plane. To my mind it seems to be the same plane itself. But when added with another plane it seems like it might be the equation of all planes passing through the intersection of the two planes. But I am not able to understand why?

fwd
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Krishan Sai
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    If $d'\neq 0$, your plane $E$ doesn't contain the origin. Then your imagination of $\lambda E=E$ doesn't work anymore. Think of $E$ defined by $x-1=0$, then $E={(1,y,z):\ y,z\in \mathbb{R}}$ and $\lambda E={(\lambda,y,z): \ y,z\in \mathbb{R}}$. –  Sep 02 '21 at 14:48
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    this might help: https://mathworld.wolfram.com/SheafofPlanes.html – David Quinn Sep 02 '21 at 15:08
  • Try collecting terms in your formula. – While I Am Sep 02 '21 at 15:09

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Say the plane you are looking for (which passes through the line of intersection (say L) of $P_1$ and $P_2$) be $P_3$. Now, you want that every point on L satisfies the equation of $P_3$.

As equation of $P_3$ is $P1+\lambda P2=0$ and every point on L already satisfies $P_1=0$ and $P_2=0$, it also satisfies $P_3$. Also, you know that $P1+\lambda P2=0$ is a linear equation in variables hence, you know it's a plane. And we have shown that it contains all points on L.

Hence, we are done.

Ilovemath
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  • L surely satisfies $P1 + \lambda*P2$ but I dont understand how does it stand for the family of planes P3. What guarantees to you that a linear combo of 2 planes is another plane – Krishan Sai Sep 02 '21 at 15:18
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    We call a plane belonging to family $P_3$ if it passes through L. We know that $P_1+\lambda P_2$ is a plane and it passes through L hence, it belongs to the family $P_3$. – Ilovemath Sep 02 '21 at 15:25
  • Any plane's equation is linear in the variables and as the linear combo gives an equation linear in variables, it's a plane. – Ilovemath Sep 02 '21 at 15:29
  • Did you get it or not? – Ilovemath Sep 02 '21 at 16:16