Question

The shortest distance between the skew lines `overliner= (6hati + 2hatj + 2hatk) + t(hati - 2hatj + 2hatk)` and `overliner = (-4hati - hatk) + s(3hati - 2hatj - 2hatk)` is ______.

Options

• 9

• `40/7`

• 108

• 120

Answer 1

The shortest distance between the skew lines `overliner= (6hati + 2hatj + 2hatk) + t(hati - 2hatj + 2hatk)` and `overliner = (-4hati - hatk) + s(3hati - 2hatj - 2hatk)` is 9.

Explanation:

Step 1: Identify vectors

A point on line 1: `bara = 6hati + 2hatj + 2hatk`

A point on line 2: `bara_2 = -4hati - hatk`

Direction vector of line 1: `barb_1 = hati -2hatj + 2hatk`

Direction vector of line 2: `barb_2 = 3hati -2hatj - 2hatk`

Step 2: Vector between the points on lines 

`bara_2 - bara_1 = (-4 - 6)hati + (0 - 2)hatj + (-1 - 2)hatk = -10hati - 2hatj - 3hatk`

Step 3: Cross product `barb_1 xx barb_2`

`barb_1 xx barb_2 = |(hati, hatj, hatk),(1, -2, 2),(3, -2, -2)|`

Expand determinant:

`hati` component: (−2)(−2) − (2)(−2) = 4 + 4 = 8

`hatj` component: −[(1)(−2) − (2)(3)] = −[−2 − 6] = 8

`hatk` component: (1)(−2) − (−2)(3) = −2 + 6 = 4

So: `barb_1 xx barb_2 = 8hati + 8hatj + 4hatk`

Step 4: Magnitude of the cross product

`|barb_1 xx barb_2| = sqrt(8^2 + 8^2 + 4^2) = sqrt(64 + 64 + 16) = sqrt144 = 12`

Step 5: Dot product with vector between lines

`(barb_1 xx barb_2) * (bara_2 - bara_1) = (8)(−10) + (8)(−2) + (4)(−3) = −80 − 16 − 12 = −108`

Step 6: Shortest Distance Formula

`D = (|(barb_1 xx barb_2) * (bara_2 - bara_1)|)/(|barb_1 xx barb_2|) = (|-108|)/12 = 9`