# Vector-Valued Lines

• Jan 30th 2013, 05:55 PM
nrr5094
Vector-Valued Lines

Find the distance between the lines:

r1(t) = (1,2,3) + t(0,-1,2)
r2(t) = (0,1,1) + t(1,1,0)

So far all I've been able to figure out is conversions to unit vector i, j, and k:

r1(t) = 0i + 1 -j +2 +2k +3 = -j + 2k + 6
r2(t) = i + j + 1

• Jan 30th 2013, 06:01 PM
chiro
Re: Vector-Valued Lines
Hey nrr5094.

You need to find a vector which is perpendicular to both lines (Hint: think about the cross product of the directional vectors of each line)
• Jan 30th 2013, 06:36 PM
Plato
Re: Vector-Valued Lines
Quote:

Originally Posted by nrr5094
Find the distance between the lines:
r1(t) = (1,2,3) + t(0,-1,2)
r2(t) = (0,1,1) + t(1,1,0)

If the two lines are are parallel chose a point on one of them and then find the distance of it to the other line.

In general, let say two lines $\displaystyle \left\{ {\begin{array}{*{20}c} {\ell _1 :P + tD_1 } \\ {\ell _2 :Q + tD_2 } \\ \end{array} } \right.$ then$\displaystyle D\left( {\ell _1 ,\ell _2 } \right) = \frac{{\left| {\left( {P - Q} \right) \cdot \left( {D_1 \times D_2 } \right)} \right|}}{{\left\| {D_1 \times D_2 } \right\|}}$