Intersection of a line and a plane, for what value(s) of k ?

In summary, the conversation discusses determining the intersection of a line and a plane using parametric form and substituting values into the Cartesian equation of the plane. It is noted that the line must pass through the plane and a value of k = -31/6 results in an infinite number of intersection points. The conversation concludes with the understanding that the initial vector of the line being on the plane is a key factor in determining the intersection.
  • #1
121
0
Well,

From what I understand, to determine the intersection of a line and a plane, we use parametric form of the line and substitute the values of x, y and z into the Cartesian equation of the plane, correct?

so, given the line
x = 2 + 4t
y = -1 + kt <=== note the 'k' variable
z = 5 - 3t

and the plane 7x + 6y - z - 3 = 0

What must be the value of k for no intersection point, one intersection point, an infinite number of intersection points?


*** My calculations this far are written below. So far I think I'm headed completely in the wrong direction but I've exhausted the only way I thought possible.

If I plug the parametric form into the plane equation, I end up with
31t + 6kt = 0

this is where I'm lost. The question has me lost, because I have not tried this form (usually I have been given a non-variable value for the parametric equations, plugging them in is easy and I go from there)

At this point, I'm -guessing- that to have no intersection point, I must have a constant on the other side of the equation and be inconsistent (of the sort, 0t = 123). This is not possible as I can't just invent one? So, there cannot be NO intersection (it must pass through the plane)

To have one intersection point, we can suppose any value of k, then t = 0?

And for infinite number of intersection points, basically I need 0t = 0 (dependent system)

SO, by letting k = - 31/6
31t + 6kt = 0
0t = 0
and thus an infinite number of intersection points (the line is contained in the plane)


*** The above is probably incorrect ( I feel) but I do not know another way. Please suggest the correct way of going about this. Do not give me the answer, I would rather just some advice on how to work my way back on this question! :smile:
 
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  • #2
Your line is
[tex]\left( {\begin{array}{*{20}c}
x \\
y \\
z \\

\end{array}} \right) = \left( {\begin{array}{*{20}c}
2 \\
{ - 1} \\
5 \\
\end{array}} \right) + t\left( {\begin{array}{*{20}c}
4 \\
k \\
{ - 3} \\
\end{array}} \right)[/tex]

Note that the point (2,-1,5) lies in the plane, which is why you can't find a value of k for which there is no intersection.

For infinite intersections, the line has to be in the plane and k = -31/6 is correct there.
 
  • #3
aha!

sorry, I'm slow on the uptake :(

thanks for the help :D
 
  • #4
You do understand now, I hope? If there's something unclear, don't hesitate to ask!

Glad that I could help :smile:
 
  • #5
Yeah, unfortunately I didn't think to look at the initial vector of the line being on the plane. I should have noticed that when I expanded the substitution and it came out to zero with the t variable left over... LOL

thanks again
 

1. What is the intersection of a line and a plane?

The intersection of a line and a plane is the point or set of points where the line and plane meet or cross each other.

2. How can the intersection be determined for a given line and plane?

The intersection can be determined by solving the equations of the line and plane simultaneously. This will give the coordinates of the point of intersection, or the equation of the line or plane if they are parallel.

3. Is there always a unique intersection for any given line and plane?

No, there are three possible outcomes when determining the intersection of a line and a plane: a unique point of intersection, no intersection (if the line and plane are parallel), or infinitely many points of intersection (if the line lies on the plane).

4. How does the value of k affect the intersection of a line and a plane?

The value of k is typically used as a variable in the equations of the line and plane. Depending on the specific equations, the value of k can determine the type and number of intersections between the line and plane.

5. Can the intersection of a line and a plane be in 3D space?

Yes, the intersection of a line and a plane can exist in any dimension, including 3D space. In this case, the intersection would be a point or line where the line and plane meet in 3D space.

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