Question

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

 

Answer 1

Equation of lines are.,

`bar r = (4 bar"i"-bar "j") + lambda(bar"i" +2bar"j" -3bar"k")` & 

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

∴ above lines passes through

`bar"a"_1 = (4 bar"i" - bar"j") "and" bar"a"_2 = (bar "i" - bar"j" + 2bar"k")`

and parallel to

`bar"b"_1 = bar"i" + 2bar"j" - 3bar"k"  &  bar"b"_2 = bar"i" - 4bar"j" -  5bar"k"`

Shortest distance =`|((bar"a"_2 - bar"a"_1).(bar"b"_1 xx bar"b"_2))/|(bar"b"_1 xx bar"b"_2)||`

`=> bar"a"_2 - bar"a"_1 = -3bar"j" + 2bar"k"`

`bar"b"_1 xx bar"b"_2 = |(bar"i",bar"j" , bar"k"),(1,2,-3),(1,4,-5)| = 2bar"i" +2bar"j" + 2bar"k"`

`therefore |bar"b"_1 xx bar"b"_2| = 2sqrt3`

Shortest distance = `|((-3bar"i"+2bar"k").(2bar"i" + 2bar"j" + 2bar"k"))/(2sqrt3)|`

`= |(-6+4)/(2sqrt3)|`

`= |-2/(2sqrt3)|`

`"d" = 1/sqrt3 "units"`