# How do I find a plane that contains two given lines?

• Mauve
In summary, the conversation discusses finding the point of intersection and equation of a plane for two given lines. The point of intersection is correctly found to be (4,0,2), but there is an error in finding the equation of the plane. The individual realizes their arithmetic mistake with the cross product and thanks the other person for pointing it out.
Mauve

## Homework Statement

a. Find a point at where these lines intersect
b. Find the equation of a plane that contains the two lines.

## Homework Equations

r[/B] = <1,3,0> + t<3,-3,2>
r = <4,0,2> + s<-3,3,0>

## The Attempt at a Solution

I correctly found the point of intersection to be (4,0,2) for part a, but for some reason, I can't find the equation of the plane. I solved the cross product for <3,-3,2> and <-3,3,0> to get <-6,-6,18> and used that to find -6(x – 4) - 6y + 18(z – 2) = 0 as my equation, but that was marked incorrect. Is this an arithmetic issue or did I approach the problem the wrong way? I've gone over it numerous times and can't find the error.

Orodruin said:
So apparently 9 - 9 = 0 and not 18 and also I am a raging imbecile. Thank you very much for your input. It really did help

Mauve said:
So apparently 9 - 9 = 0 and not 18 and also I am a raging imbecile.
Arithmetic error happens to everyone and when your brain has told you one thing once it may be difficult to find it until someone points it out.

Mark44

## 1. What is the equation for a plane that contains two given lines?

The equation for a plane that contains two given lines can be found by taking the cross product of the direction vectors of the two lines. This will give you the coefficients for the x, y, and z variables in the equation.

## 2. How do I determine if two lines are parallel or intersecting?

Two lines are parallel if their direction vectors are parallel (i.e. have the same slope). Two lines are intersecting if their direction vectors are not parallel.

## 3. Can I find the equation for a plane if the lines are skew?

No, if two lines are skew (not parallel or intersecting), then there is no plane that contains both lines. In this case, you would need to find the equation for two separate planes, each containing one of the lines.

## 4. What information do I need to find the equation for a plane containing two lines?

You will need the direction vectors of the two lines, as well as a point that lies on each line. This can be used to find the normal vector for the plane, which is necessary for writing the equation.

## 5. Are there any special cases when finding the equation for a plane containing two lines?

Yes, if the two lines are parallel, then there are infinitely many planes that contain both lines. In this case, any of these planes can be represented by the same equation. Additionally, if the two lines are coincident (i.e. they are the same line), then there are also infinitely many planes that contain them, and any equation for the line can be used as the equation for the plane.

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