The equation of a line in 3space

  • #1

Homework Statement

Find parametric equations of a line that intersect both L1 and L2 at right angles.
L1: [x,y,z]=[4,8,-1] + t[2,3,-4] L2: (x-7)/-6 = (y-2)/1 = (z+1)/2

Homework Equations

Symmetric, parametric, cross product

The Attempt at a Solution

We used cross product to find the direction vector, which was [1,2,2]. To finish the parametric equation, we need at least one of the points that this vector intersects the two lines (which are skewed).

One attempt was setting the two line's parametric equations equal to eachother to isolate the t or s, and using this to find the x-, y- and z-value, this should not be possible though because the lines are skewed -- the t and s should not be equal to eachother... But the numbers do satisfy eachother (but not the back of the book -- the books example answer of a point (2,5,3) does not work).

We then tried taking the 's' we found, and put it into the parametric equations set to eachother, to find the 't' at the set 's'. This resulted in -3/10, -1, -3/10. We then used these 't' values to find the x-, y- and z- values, and eventually make a parametric equation. This also does not work when we check it.

Answers and Replies

  • #2
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Write both in parametric form. Let f = the distance squared between a point on one line and a point on the other. That will be a function of s and t. Minimize it as a function of s and t. Those values of s and t will give you the closest points and you can go from there.