Question

Find the shortest distance between the lines, `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr = - 4hati - hatk + mu(3hati - 2hatj - 2hatk)`

Options

• 6

• 7

• 8

• 9

Answer 1

9

Explanation:

If `veca_1` and `vecb_2` be the two points on the lines and `vecb_1` and `vecb_2` be their direction, the shortest distance between them.

`|((veca_1 - veca_2) * (vecb_1 xx vecb_2))/|vecb_1 xx vecb_2||`

The given lines are,

`vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` 

And `vecr = 4hati +- hatk + mu(3hati - 2hatj - 2hatk)`

Here, `veca_1 = 6hati - 2hatj + 2hatk, veca_2 = - 4hati - hatk`

 `vecb_1 = hati - 2hatj + 2hatk, vecb_2 = 3hati - 2hatj - 2hatk`

`veca_1 - veca_2 = (6hati - 2hatj + 2hatk) - (-4hati - hatk = 10hati + 2hatj + 3hatk)`

`vecb_1 xx vecb_2 = |(hati, hatj, hatk),(1, -2, 2),(3, -2, -2)| = 8hati + 8hatj + 4hatk`

`|vecb_1 xx vecb_2| = sqrt(8^2 + 8^2 + 4^2)` = 12

S.D. = `((veca_1 - veca_2) * (vecb_1 xx vecb_2))/|b_1 xx b_2|`

= `((10hati + 2hatj + 3hatk) * (8hati + 8hatj + 4hatk))/12`

= `(80 + 16 + 12)/12`

= `108/12`

= 9