How to determine if two line intersects each other?!
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How to determine if two line intersects each other?!

[From: ] [author: ] [Date: 11-05-11] [Hit: ]
2] + t[1, 8,L2: [x,y,z]= [2, -19,......
Determine if line L1 intersects line L2.

L1: [x,y,z]= [4, -3, 2] + t[1, 8, -3]
L2: [x,y,z]= [2, -19, 8] + s[4, -5, -9]

Please provide a theoretical solution on how to figure it out, I know how to find it graphically. Thanks for your time!

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if the lines intersect, there's a value of t and a value of s for which they both produce the same values of x, y, and z. we solve for s and t.

4 + t = x = 2 + 4s → 4 + t = 2 + 4s → 4s - t = 2
-3 + 8t = y = -19 - 5s → -3 + 8t = -19 - 5s → 5s + 8t = -16

5s + 8t = -16
32s - 8t = 16
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37s ∙ ∙ ∙ = 0
∙ ∙ ∙ ∙ s = 0 ∙ ∙ ∙ ∙ ∙ and so t = -2

check it:
[x,y,z] = [2,-19,8] + 0[4,-5,-9] = [2,-19,8] and
[x,y,z] = [4,-3,2] - 2[1,8,-3] = [2,-19,8]

so yes, the lines intersect.

note: David ignored the possibility of skew lines which are not parallel yet don't intersect.

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If these two lines intersect, you will have three equations in two unknowns. Pick two equations and solve for t and s. Then see if your solution also holds for the third equation. If it does the lines intersect.

4 + t = 2 + 4s

-3 + 8t = -19 -5s

s = 0 and t = -2

Now go the the third equation

2 - 3t = 8 - 9s

Notice that s = 0 and t = -2 works in the third equation.

The point of intersection is [2,-19,8] + 0[4,-5,-9] = [2,-19,8]

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The lines intersect if and only if they are not parallel.

The lines are parallel if and only if their direction vectors are parallel.

Since [1, 8, -3] is not a multiple of [4, -5, -9] (by multiple I mean multiplied by any real number), we determine that the direction vectors are not parallel.

So the lines intersect.
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