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Determine the equation of the plane that passes through the point of intersection of the planes $$P=\begin{cases} 2x + z - 7 = 0\\ x - y = 0\\ x + y - 2z + 2=0 \end{cases}$$ and passes through the origin and the point $Q=(2, 3, 8)$. I tried a lot to find the solution of this problem but in vain.

ecrin
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Mohamed Asem
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  • Welcome to MSE. Here's how to ask a good question. Follow these guidelines to get help in this forum. It's particularly important that you share your own work and thoughts on the problem to show that you have made a serious effort by yourself before asking for help, and you're not just trying to get others to solve it for you. This is not a homework service. – jjagmath Mar 27 '23 at 13:35
  • I searched for the way to solve such that problem in a lot of sites and books but I didn't find the answer for my question. – Mohamed Asem Mar 27 '23 at 13:43
  • Do you mean the intersection point of the three planes? – Mohamed Asem Mar 27 '23 at 13:52
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    Ok, P=(2, 2, 3) what should I do after that? – Mohamed Asem Mar 27 '23 at 13:58
  • It is Ax + By + Cz + D =0 where A, B, C are direction ratios perpendicular to the plane and D is the length of the column falling form the origin to the plane. – Mohamed Asem Mar 27 '23 at 14:12
  • These points must achieve the equation of the plane that means by substituting for these points in the equation of a plane passing through three points we get the equation of the required plane. – Mohamed Asem Mar 27 '23 at 14:23
  • Well, do that and tell us the 3 equations on $a,b,c,d$ you obtain. Then, solve this system. You will find $d=0$ and (for instance) $b=$ a certain number times $c$ and $a=$ another number times $c.$ – Anne Bauval Mar 27 '23 at 14:28
  • The three equations are:(1) 2A + 2B + 3C=0 (2) Ax + By + Cz=0 (3) 2A + 3B + 8C=0 – Mohamed Asem Mar 27 '23 at 14:30
  • I agree with (1) and (3) but not with (2), I don't know where you took it from. Before (1) and (3), you seem to have noticed that $d=0$ is the condition for $O$ to belong to the plane. So, our 3 equations are (0): $d=0$ and your (1) and (3). Now, solve [(1)and(3)] like I hinted before. – Anne Bauval Mar 27 '23 at 14:34
  • Sorry I mean that the three equations are:(1) 2A + 2B + 3C + D=0 (2) Ax + By + Cz=0 (3) 2A + 3B + 8C + D=0 by solving the three equation we get the equation of the required plane: 7x - 10y + 2z=0 – Mohamed Asem Mar 27 '23 at 14:41
  • Your equation (2) makes no sense. There must be no $x,y,z.$ How do you express that $O$ belongs to the plane? And how did you find "the required plane" since you do not have the correct equations? – Anne Bauval Mar 27 '23 at 14:43
  • That means that D=0 so the plane passes through the origin in this case as it is khown. – Mohamed Asem Mar 27 '23 at 14:45
  • Let us continue this discussion in chat. – Anne Bauval Mar 27 '23 at 14:46
  • Here the beginning of a chain of 3 duplicates, selected in this list. – Anne Bauval Mar 27 '23 at 16:04

2 Answers2

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Here is a quick way of solving this.

The required plane must take the form $$2x+z-7+\lambda(x-y)+\mu(x+y-2z+2)=0$$ For some $\lambda, \mu\in\mathbb{R}$

This is because it is the equation of a plane, and it is satisfied by the point which simultaneously satisfies the equations of all three given planes (i.e. the point of intersection).

Now $(0,0,0)$ lies on the plane, so $\mu=\frac72$.

Also $(2,3,8)$ lies on the plane, so $\lambda=-\frac{53}{2}$

Using these values of $\lambda$ and $\mu$ gives the equation of the plane, which simplifies to $$-7x+10y-2z=0$$

David Quinn
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This is the rule used for finding the equation of any plane passing through three coplinear points.

Mohamed Asem
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  • I am disappointed: this is not what you planned to do in the chat (and which would have been quicker). But never mind, complete the calculations in this answer. I.e. replace the $x_i,y_i,z_i$ by their values and compute this determinant. – Anne Bauval Mar 27 '23 at 14:57
  • I tried to show you the way I used for solving the three equation in a method or another and there are no difference in the final result so any method I mentioned it is right. – Mohamed Asem Mar 27 '23 at 15:00
  • I know that method but I tried to teach you a quicker one in the chat, and it was almost finished (you nearly had the final the result with that method), whereas here there still is much work to do. – Anne Bauval Mar 27 '23 at 15:05
  • Oh I have understood everything, thank you sir. – Mohamed Asem Mar 27 '23 at 15:32
  • It would be good for you to present your end of the resolution, by both methods. – Anne Bauval Mar 27 '23 at 15:41