Show that L_1 and L_2 intersect, and find there point of intersection.?

Question

L_1: x=2+t, y=2+3t, z=3+t
L_2: x=2+t, y=3+4t, z=4+2t

Answer 1

One thing that makes this easy is that x=2+t for both lines. That means that if L_1 and L_2 intersect, t must be the same for both lines at the point of intersection.

So, y=2+3t=3+4t
2+3t=3+4t
2-3=4t-3t
-1=t

Check for z:
3+t(=?)4+2t, where t=-1
3+(-1) (=?) 4+2(-1)
3-1 (=?) 4-2
2 (=) 2

So, the lines do intersect, at the point where t=-1, or
(2+(-1), 2+3(-1), 3+(-1)), or
(2-1, 2+(-3), 3-1), or
(1, -1, 2)

Answer 2

If they intersect, then they have a point in common. The two "x" expressions are the same, so skip ahead to the "y" expressions. Set them equal to each other and solve for t. if the same t value gives equal z values back, then the two lines have a point in common.

Answer 3

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