# Thread: Show that the two lines do not intersect.

1. ## Show that the two lines do not intersect.

Hi all working through Anton and Busby.., if i have two vectors, v = (1,2,1)s and v = (9,6,0) + (0,1,-1)t how can i show that they do not intersect in 3-space ?

2. By trying to find a point where they intersect! If these two lines intersected at (x, y, z), then we would have (x, y, z)= (s, 2s, s)= (9, 6+ t, -t) so we have s= 9, 2s= 6+ t, s= -t. Since the first equation tells you what s is, the third equation tells you what t must be. Do those values satisfy the second equation.

3. HallsofIvy, thanks. it simplifies to 18 = -3. In this case would it be possible to find a line orthogonal to both lines? if so how can I find this. Thanks.

4. Actually i think an orthogonal vector would just be n = (1,2,1)*(0,1,-1) = (0,2,-1). Would the vector then be x = (9,6,0) + (0,2,-1)t or would i need to omit the first point (9,6,0) ?

5. The orthogonal to two vectors is cross product
$
n = (1,2,1) \; \times \; (0,1,-1)=(-3,1,1).
$

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### show that the line do not intersect each other

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