Question

By computing the shortest distance, determine whether following lines intersect each other.

`bar"r" = (hat"i" - hat"j") + lambda(2hat"i" + hat"k") and bar"r" = (2hat"i" - hat"j") + mu(hat"i" + hat"j" - hat"k")`

Answer 1

The shortest distance between the lines
`bar"r" = bar"a"_1 + lambdabar"b"_1 and bar"r" = bar"a"_2 + mubar"b"_2` is given by 

Here, `bar"a"_1 = hat"i" - hat"j", bar"a"_2 = 2hat"i" - hat"j", bar"b"_1 =2hat"i" + hat"k", bar"b"_2 = hat"i" + hat"j" - hat"k"`.

`bara_2 - bara_1 = (2hati-hatj) - (hati-hatj)`

∴ `bara_2 - bara_1 = hati`

∴ `bar"b"_1 xx bar"b"_2 = |(hat"i",hatj, hat"k"),(2, 0, 1),(1, 1, -1)|`

= `(0 - 1)hat"i" - (-2 - 1)hat"j" + (2 - 0)hat"k"`
= `-hat"i" + 3hat"j" + 2hat"k"`
and
∴ `(bar"a"_2 - bar"a"_1).(bar"b"_1 xx bar"b"_2) = hat"i".(-hat"i" + 3hat"j" + 2hat"k")`
= 1(– 1) + 0(3) + 0(2) 
= – 1
and
`|bar"b"_1 xx bar"b"_2| = sqrt((-1)^2 + 3^2 + 2^2)`
= `sqrt(1 + 9 + 4)`
= `sqrt(14)`

d = `|((bar"a"_2 - bar"a"_1).(bar"b"_1 xx bar"b"_2))/|bar"b"_1 xx bar"b"_2||`.
∴ the shortest distance between the given lines
= `|(-1)/sqrt(14)|`

= `(1)/sqrt(14)"unit"`
Hence, the given line do not intersect.