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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.
Solution
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.
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