Three Dimensional Geometry

Find the shortest distance between the lines: and - Wired Faculty

Question

Find the shortest distance between the lines:
$straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space straight k with hat on top right parenthesis$
and $straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight mu left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis$

Solution

The equations of two lines are

straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight mu left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis

Comparing these equations with

straight r with rightwards arrow on top space equals space stack straight a subscript 1 with rightwards arrow on top space plus space straight lambda space stack straight b subscript 1 with rightwards arrow on top space space and space straight r with rightwards arrow on top space equals stack straight a subscript 2 with rightwards arrow on top space plus space straight mu space stack straight b subscript 2 with rightwards arrow on top comma

we get

stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top comma space space stack straight b subscript 1 with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight k with hat on top comma space space stack straight a subscript 2 with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top comma space space stack straight b subscript 2 with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top

stack straight a subscript 2 with rightwards arrow on top space minus space straight a with rightwards arrow on top space equals space straight i with hat on top

stack straight b subscript 1 with rightwards arrow on top space cross times rightwards arrow