Write the vector equation of the following lines and hence find the shortest distance between them : (x - 1)/2 = (y - 2)/3 = (z - 3)/4

Question

Write the vector equation of the following lines and hence find the shortest distance between them :

\(\cfrac{\text x-1}2=\cfrac{y-2}3 = \cfrac{z-3}4\) and \(\cfrac{\text x-2}3=\cfrac{y-3}4=\cfrac{z-5}5\)

(x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 2)/3 = (y - 3)/4 = (z - 5)/5

Answer 1

To Find : i) vector equations of given lines

ii) distance d

Formulae :

1. Equation of line :

Equation of line passing through point A (a₁, a₂, a₃) and having direction ratios (b₁, b₂, b₃) is \(\vec r=\bar a_1+\lambda \bar{b}\)

then,

4. Shortest distance between two lines :

The shortest distance between the skew lines \(\vec r=\bar a_1+\lambda \bar{b}\) and 

\(\vec r=\bar a_2+\lambda \bar {b}\) is given by,

Given Cartesian equations of lines

L1 : \(\cfrac{\text x-1}2=\cfrac{y-2}3 = \cfrac{z-3}4\) 

Line L1 is passing through point (1, 2, 3) and has direction ratios (2, 3, 4)

Therefore, vector equation of line L1 is

And

L2 : \(\cfrac{\text x-2}3=\cfrac{y-3}4=\cfrac{z-5}5\)

Line L2 is passing through point (2, 3, 5) and has direction ratios (3, 4, 5)

Therefore, vector equation of line L2 is

Now,

= - 2 + 2 - 2

= -2

Therefore, the shortest distance between the given lines is