Angle between lines, with free variables in equations?

In summary: The directions are (1,-3,4) and (-2,3,2). The dot product is -3, so cos(\theta) = -3/(sqrt(26)sqrt(17)). In summary, to find the cosine of the angle formed by the intersecting lines m1 and m2, you can use the dot product formula a⋅b = length(a)length(b) cosθ, where a and b are the direction vectors of the lines. In this case, the cosine of the angle is equal to -3 divided by the product of the lengths of the direction vectors.
  • #1
Oliviacarone
18
1

Homework Statement


Find the cosine of the angle determined by the intersecting lines
m1: (x,y,z) = (-2,1,4)+s(1,-3,4)
m2: (x,y,z) = (-2,1,4)+t(-2,3,2)

s and t are free variables

Homework Equations


a⋅b = length(a)length(b) cosθ

The Attempt at a Solution


I just did this equation using no free variables, since I have no idea how to use them and someone told me to ignore them? So I did
((-2,1,4)⋅(-2,1,4))/(√(-2+1+4))((√(-2+1+4)) = cosθ and solved for θ.
 
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  • #2
Oliviacarone said:

Homework Statement


Find the cosine of the angle determined by the intersecting lines
m1: (x,y,z) = (-2,1,4)+s(1,-3,4)
m2: (x,y,z) = (-2,1,4)+t(-2,3,2)

s and t are free variables

Homework Equations


a⋅b = length(a)length(b) cosθ

The Attempt at a Solution


I just did this equation using no free variables, since I have no idea how to use them and someone told me to ignore them? So I did
((-2,1,4)⋅(-2,1,4))/(√(-2+1+4))((√(-2+1+4)) = cosθ and solved for θ.
What you have here are parametric representations of your two lines. That means that different values of the parameter s give you different points on line m1, for example.

So if s = 1, then (x,y,z) = (-2,1,4)+1(1,-3,4) = (-1,-2,0).

If s = 0, then (x,y,z) = (-2,1,4)

If you want to find the angle formed by the two lines, the first thing you need to do is figure out at what point they intersect. That means you need to find a value of s and a value of t that give the same point (x,y,z). That shouldn't take you very long.

Once you know that, you can find the angle between the two lines using the equation you have written in the Relevant Equations section, but you need to substitute the correct vectors for a and b. Now that you know what s and t mean, that should not be difficult either.

Give it try!
 
  • #3
tnich said:
What you have here are parametric representations of your two lines. That means that different values of the parameter s give you different points on line m1, for example.

So if s = 1, then (x,y,z) = (-2,1,4)+1(1,-3,4) = (-1,-2,0).

If s = 0, then (x,y,z) = (-2,1,4)

If you want to find the angle formed by the two lines, the first thing you need to do is figure out at what point they intersect. That means you need to find a value of s and a value of t that give the same point (x,y,z). That shouldn't take you very long.

Once you know that, you can find the angle between the two lines using the equation you have written in the Relevant Equations section, but you need to substitute the correct vectors for a and b. Now that you know what s and t mean, that should not be difficult either.

Give it try!
Oh, one more thing - you need to calculate the lengths of a and b correctly.
 
  • #4
tnich said:
What you have here are parametric representations of your two lines. That means that different values of the parameter s give you different points on line m1, for example.

So if s = 1, then (x,y,z) = (-2,1,4)+1(1,-3,4) = (-1,-2,0).

If s = 0, then (x,y,z) = (-2,1,4)

If you want to find the angle formed by the two lines, the first thing you need to do is figure out at what point they intersect. That means you need to find a value of s and a value of t that give the same point (x,y,z). That shouldn't take you very long.

Once you know that, you can find the angle between the two lines using the equation you have written in the Relevant Equations section, but you need to substitute the correct vectors for a and b. Now that you know what s and t mean, that should not be difficult either.

Give it try!
Hmm should I just try s=0 and t=0? Not sure what else I would use.
 
  • #5
Oliviacarone said:
Hmm should I just try s=0 and t=0? Not sure what else I would use.
I don't think you have a clear picture of the problem. Try drawing it in two dimensions. So plot y vs. x for m1 and m2. Let s and t go from -1 to 1. Look at the angle that forms.
 
  • #6
Oliviacarone said:

Homework Statement


Find the cosine of the angle determined by the intersecting lines
m1: (x,y,z) = (-2,1,4)+s(1,-3,4)
m2: (x,y,z) = (-2,1,4)+t(-2,3,2)

s and t are free variables

Homework Equations


a⋅b = length(a)length(b) cosθ

The Attempt at a Solution


I just did this equation using no free variables, since I have no idea how to use them and someone told me to ignore them? So I did
((-2,1,4)⋅(-2,1,4))/(√(-2+1+4))((√(-2+1+4)) = cosθ and solved for θ.
m1 and m2 start at the same point (-2,1,4). The directions are (1,-3,4) and (-2,3,2). The dot product is -3, so [tex]cos(\theta )=\frac{-3}{\sqrt{442}}[/tex].
 
  • #7
@Oliviacarone, please post homework questions in one of the forum sections under Homework & Coursework, not in the technical math sections. I think this is the second one of yours I have moved today.
 

1. What is the angle between two lines with equations that have free variables?

The angle between two lines can be determined by finding the inverse tangent of the slopes of the lines. If the equations of the lines have free variables, the slopes can be substituted with the values of the variables to find the angle.

2. Can the angle between two lines change if the equations have different free variables?

Yes, the angle between two lines can change if the equations have different free variables. This is because the slopes of the lines can change depending on the values of the variables, thus affecting the angle between them.

3. Is it possible for the angle between two lines to be negative if the equations have free variables?

Yes, the angle between two lines can be negative if the equations have free variables. The angle is measured counterclockwise from the first line to the second line, so if the angle is greater than 180 degrees, it will be represented as a negative angle.

4. How can the angle between two lines be used in real-world applications?

The angle between two lines can be used in various fields such as engineering, physics, and architecture. It can be used to determine the direction and orientation of objects, calculate distances, and solve problems involving intersecting lines.

5. Can the angle between two lines be greater than 360 degrees?

No, the angle between two lines cannot be greater than 360 degrees. The angle is measured within a circle, and a full circle is 360 degrees. If the angle between two lines is greater than 360 degrees, it means that the lines are not intersecting and are parallel to each other.

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