Question

The shortest distance between the lines

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

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

पर्याय

• `3sqrt2`

• `3/sqrt2`

• `5/sqrt2`

• `5sqrt2`

Answer 1

`3/sqrt2`

Explanation:

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

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

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

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

Now, `bar"a"_2 - bar"a"_1 = hat"i" - 3hat"j" - 2hat"k"`

and `bar"b"_1 xx bar"b"_2 = |(hat"i",hat"j", hat"k"),(1,-1,1),(2,1,2)|`

`= hat"i" (- 2 - 1) - hat"j"(2 - 2) + hat"k"(1 + 2)`

`= -3hat"i" + 3hat"k"`

∴ `(bar"a"_2 - bar"a"_1)(bar"b"_1 xx bar"b"_2) = (hat"i" - 3hat"j" - 2hat"k")*(-3hat"i" + 3hat"k")`

= 1(- 3) - 3(0) - 2(3) = - 9

`|bar"b"_1 xx bar"b"_2| = sqrt((-3)^2 + 0^2 + 3^2)`

`= sqrt18 = 3sqrt2`

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

`= |(-9)/(3sqrt2)| = 3/sqrt2` units