Equation of line parallel to plane and intersaction with other line

In summary, the problem is to find the equation of a line passing through point M(1,0,7), parallel to the plane 3x-y+2z-15=0, and intersecting the line \frac{x-1}{4}=\frac{y-3}{2}=\frac{z}{1}. The equation of the line will be \frac{x-1}{a_1}=\frac{y}{a_2}=\frac{z-7}{a_3}, and we need to find \vec{a}(a_1,a_2,a_3) and three conditions in the system. The first condition is \vec{a} \circ \vec{n}=0
  • #1
Theofilius
86
0

Homework Statement



Hello!

I have one problem which seems not so difficult:

-Find the equation of line which passes throught the point M(1,0,7), parallel of the plane 3x-y+2z-15=0 and it intersects the line [tex]\frac{x-1}{4}=\frac{y-3}{2}=\frac{z}{1}[/tex]

Homework Equations





The Attempt at a Solution



The equation of the line will be: [tex]\frac{x-1}{a_1}=\frac{y}{a_2}=\frac{z-7}{a_3}[/tex]

So we need to find [tex]\vec{a}(a_1,a_2,a_3)[/tex] and we need three conditions in the system.

The first condition is [tex]\vec{a} \circ \vec{n}=0[/tex] or [tex](a_1,a_2,a_3)(3,-1,2)=0[/tex] or [tex]3a_1-a_2+2a_3=0[/tex].

The second condition is the intersection of two lines, and it is:

[tex]-17a_1+28a_2+12a_3=0[/tex]

What about the third condition?
 
Physics news on Phys.org
  • #2
There is no third condition. You cannot determine a1, a2, and a3 uniquely. Any multiple of a given a1, a2, and a3 will also determine the same line. From the two equations you have you can solve for two of the numbers as functions of the third. Choose that third as you please, say equal to 1, and solve for the other 2.
 
  • #3
Other way (I think it is very similar to what you did):
r = (1,3,0) + t (4,2,1)
P = (1,0,7)
find direction vector: r - P
and you know n.(r-P) = 0

So, only one condition .. (and only one unknown)
 
  • #4
rootX what is r, and what is P?
 
  • #5
Physicsissuef said:
rootX what is r, and what is P?

r is (1,3,0) + t (4,2,1) ... a line
P is (1,0,7) .. a point

:smile:

Should work, shouldn't it?
 
  • #6
It wouldn't work, since by your opinion you will find point, product of the intersection of line and plane... And we'll need to find parallel vector to the line which satisfies the above conditions...
 

What is the equation of a line parallel to a plane?

The equation of a line parallel to a plane can be written as ax + by + cz = d, where a, b, and c are the coefficients of the line's direction vector and d is a constant.

How is the equation of a line parallel to a plane related to the normal vector of the plane?

The normal vector of a plane is perpendicular to the plane and is used to determine the coefficients a, b, and c in the equation of a line parallel to that plane. The direction vector of the line will be parallel to the normal vector of the plane.

What is the intersection point of a line with a plane?

The intersection point of a line with a plane is the point where the line crosses or touches the plane. This point can be found by solving the equations of the line and the plane simultaneously.

How many intersection points can a line and a plane have?

A line and a plane can have either zero, one, or infinite intersection points. This depends on the orientation and position of the line and the plane relative to each other. For example, if the line is parallel to the plane, they will not have any intersection points.

How can we determine if two lines are parallel?

Two lines are parallel if their direction vectors are parallel. This means that the coefficients a, b, and c in their equations are proportional to each other. If a/a = b/b = c/c, then the lines are parallel.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
7
Views
520
  • Precalculus Mathematics Homework Help
Replies
17
Views
907
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
6
Views
161
  • Precalculus Mathematics Homework Help
Replies
17
Views
1K
  • Precalculus Mathematics Homework Help
Replies
9
Views
2K
Replies
1
Views
523
  • Precalculus Mathematics Homework Help
Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
3
Views
2K
  • Precalculus Mathematics Homework Help
Replies
6
Views
4K
Back
Top