Angle between lines, with free variables in equations?

[Oliviacarone]

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.

 

[tnich]

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!

 

[tnich]

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.

 

[Oliviacarone]

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.

 

[tnich]

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.

 

[mathman]

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 cos(\theta )=\frac{-3}{\sqrt{442}}.

 

[Mark44]

@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.