Find the shortest distance between the following lines : (x - 3)/1 = (y - 5)/(-2) = (z - 7)/1

Question

Find the shortest distance between the following lines : \(\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}\) and \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}\)

Answer 1

Let \(\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}=\lambda\) and \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}=k\)

Now, let's take a point on first line as

A (λ + 3, -2λ + 5, λ + 7) and let

B (7k - 1, -6k - 1, k-1) be point on the second line

The direction ratio of the line AB

Now, as AB is the shortest distance between line 1 and line 2 so,

Solving equation (i) and (ii), we get

λ = 0 and k = 0