Finding equations of two lines

[mrroboto]

Homework Statement

Find 2 lines in R3 that are not parallel and do not intersect.

Homework Equations

orthogonal: a.b=0
parallel: a=tb

The Attempt at a Solution

(1,1,1) and (2,3,4)

(1,1,1).(2,3,4) = 7
and there does not exist a t such that
t(1,1,1) = (2,3,4)

 

[Defennder]

Is that all you're given? Any 2 lines would do?

Your answer doesn't include the starting point of the line vector it's supposed to be in: (x0,y0,z0) + t(a,b,c). The point (x0,y0,z0) isn't specified. That's why you haven't shown why they do not intersect.

 

[HallsofIvy]

If fact, your answers are not lines at all! What you give look like points but I guess you mean them as vectors- and I presume you mean them as "direction" vectors for the lines. The fact that "(1,1,1).(2,3,4) = 7" shows that they are not perpendicular (not really relevant) and the fact that "there does not exist a t such that t(1,1,1) = (2,3,4)" shows that they are not parallel. Neither of those means they do not intersect.
As defennnder said, you need to write them as lines with a parameter such as t. To do that, you will have to specify a point on them- and whether or not they intersect will depend on that point. For example, the lines $\vec{r_1}= t\vec{i}+ t\vec{j}+ t\vec{k}$ and $\vec{r_2}= 2t\vec{i}+ 3t\vec{j}+ 4t\vec{k}$ have direction vectors $\vec{i}+ \vec{j}+ \vec{k}$ and $2\vec{i}+ 3\vec{j}+ 4\vec{k}$ respectively and intersect at (0, 0, 0). You need to find skew lines.

 

[CompuChip]

First try to get the geometrical picture clear. Can you describe in words two lines that would do?