# Linear Intersections

1. Apr 29, 2010

### spoc21

1. The problem statement, all variables and given/known data

Find the vector, parametric and symmetric equations of a line that intersect both line 1 and line 2 at 90°.

L1 :
x = 4 + 2t
y = 8 + 3t
z = -1 - 4t
L2 :
x = 7 - 6t
y = 2+ t
z = -1 + 2t

2. Relevant equations

3. The attempt at a solution

I tried taking the cross product of [2,3,-4], and [-6,1,2], to find the vector perpendicular to both lines

= [10,20,20]

So our new direction vector is [10,20,20]

Now I'm stuck (confused about how to find the equations)

Any help would be greatly appreciated!

2. Apr 29, 2010

### lanedance

pick an arbitrary point on each line and write the equation of the line formed by joing the point, in terms of the points (use different parameteric variables for each line, say s & t)

then use the direction you have found to solve for s & t

3. May 1, 2010

### spoc21

Thanks lanedance,
could you elaborate a little more. what points would I need to pick, in order to solve for the equations.

Thanks,

4. May 2, 2010

### lanedance

you need to solve for the points that give you the connecting line, which is perpindicular to L1 & L2.

The first point, say p1, is a point on L1, the 2nd, p2 on L2. The direction of p1,p2 must be parallel or anti parallel to the direction you found. Use that fact to solve for p1 & p2 simultaneously.