Question

By computing the shortest distance determine whether the following lines intersect each other: `(x -5)/(4) = (y - 7)/(5) = (z + 3)/(5)` and x – 6 = y – 8 = z + 2.

Answer 1

Let `bara_1` be the position vector of the point line `(x -5)/(4) = (y - 7)/(5) = (z + 3)/(5)`

`barb_1` be the vector parallel to this line

`bara_1 = 5hati + 7hatj-3hatk`

`barb_1 = 4hati+5hatj+5hatk`

Let `bara_2` be the position vector of the point line `(x  –  6)/1 = (y  –  8)/1 = (z + 2)/1`

`barb_2` be the vector parallel to this line

`bara_2 = 6hati + 8hatj-2hatk`

`barb_2 = hati+hatj+hatk`

`bara_2-bara_1=(6hati + 8hatj-2hatk) -(5hati + 7hatj-3hatk) `

`= hati+hatj+hatk`

`barb_1xxbarb_2=|(hati,hatj,hatk),(4,5,5),(1,1,1)|`

`=hatj - hatk`

`|barb_1xxbarb_2|= sqrt(1^2+(-1)^2)=sqrt(2)`

`(bara_2-bara_1).(barb_1xxbarb_2) = 0+1-1=0`

Shortest distance between the given lines

`= |((bara_2-bara_1).(barb_1xxbarb_2))/|barb_1xxbarb_2||`

`= |0/sqrt(2)|= 0`

∴ the lines intersect each other.