Question

Shortest distance between the two lines and find
two points (one on each line) which are closest together.
[1,0,−1]^T +t[1,1,1]^T and [0,1,1]^T +s[2,2,2]^T

Answer 1

@phi help!

Answer 2

these lines are parallel (same direction vector (1,1,1) )

Answer 3

can you please show me how you got the answer

Answer 4

I did not get the answer. I just pointed out that (1,1,1) and (2,2,2) point in the same direction, so the lines are parallel. we still have to figure out how far apart they are.

Answer 5

ohh i see

Answer 6

so there is no two points closest to each other

Answer 7

there are lots of points in nearest to each other. I think you need to find one pair, and the distance.

Answer 8

how do i find the distance

Answer 9

ok, I finally had a thought.

Answer 10

both lines are going in the same direction. we can use that direction vector as a normal to a plane in 3d.

For example, if we look at the plane that contains point [0,1,1] and has a normal vector (1,1,1), we have this equation
\[ P \cdot (1,1,1) = b\]
replacing P with (0,1,1) we find b is 2, and the equation of the plane is
\[ P \cdot (1,1,1) = 2\]

now we need the point on the other line that is in this same plane
Let P= \( (1,0,−1)^T +t(1,1,1)^T \)
\[ ( (1,0,−1)^T +t(1,1,1)^T) \cdot (1,1,1) = 2 \\
(1,0,−1)\cdot (1,1,1)+ t (1,1,1)\cdot (1,1,1)= 2 \\
0 + 3t=2\\
t= \frac{2}{3}
\]
that means the point on the other line is
(1,0,−1) +(⅔)(1,1,1) = (5/3 , ⅔, -⅓)