Equation of a plane containing two lines

In summary, the conversation discusses finding an equation for the plane containing two given lines and determining the correct value for C in the equation 3x - 12y + 12z = C by using the given line equations and plugging in a value for t.
  • #1
Whiz
20
0

Homework Statement



Find an equation for the plane containing the lines:
x-3= 4t , y-4 = t, z-1 = 0
x+1 = 12t, y-7 = 6t, z-5 = 3t

The Attempt at a Solution



I found the equations of both lines.
First line:
(3, 4, 1) + t(4, 1, 0)
(-1, 7, 5) + t(12, 6, 3)

Now I take the cross product of (4, 1, 0) and (12, 6, 3) to get (3, -12, 12)

I found two answers from reviewing this question and I'm confusing myself, so I was wondering which answer is right. (If any)

3x - 12y + 12z = 27
or
3x - 12y + 12z = 0

Thanks in advance.
 
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  • #2
Well, you know several points on the plane. Try a few in your equations.
 
  • #3
I don't think either one is right. You are right that the plane equation is 3x - 12y + 12z=C. Find the correct value for C by putting say t=0 into your line equations and solving for it.
 

Related to Equation of a plane containing two lines

What is the equation of a plane containing two lines?

The equation of a plane containing two lines can be written in the form Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the x, y, and z variables, and D is a constant. This equation represents the set of all points that are simultaneously on both lines.

How do you find the equation of a plane containing two lines?

To find the equation of a plane containing two lines, you can first find the direction vectors for each line. Then, use the cross product of these direction vectors to find the normal vector of the plane. Finally, plug in the coordinates of a point on one of the lines into the equation Ax + By + Cz + D = 0 and solve for D.

Can two parallel lines be contained in the same plane?

Yes, two parallel lines can be contained in the same plane. This is because parallel lines have the same direction vector, which means the cross product of their direction vectors will be zero, resulting in a normal vector of (0,0,0). This in turn means that the equation of the plane will not have a D term, resulting in a plane that contains both lines.

Is it possible for two perpendicular lines to be contained in the same plane?

Yes, two perpendicular lines can be contained in the same plane. This is because perpendicular lines have direction vectors that are orthogonal to each other, which means their cross product will result in a normal vector that is also orthogonal to both lines. This results in a plane that contains both lines.

Can a plane containing two lines intersect with another line?

Yes, a plane containing two lines can intersect with another line. This is because the intersection of a plane and a line is a single point, and the intersection of two planes is a line. So, the intersection of a plane containing two lines and another line would result in a single point or a line.

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