Find the value of λ for which the following lines are perpendicular to each other:

`(x - 5)/(5 lambda + 2 ) = ( 2 - y )/5 = (1 - z ) /-1 ; x /1 = ( y + 1/2)/(2 lambda ) = ( z -1 ) / 3`

#### Solution

`(x - 5)/(5 lambda + 2 ) = ( 2 - y )/5 = (1 - z ) /-1 ; x /1 = ( y + 1/2)/(2 lambda ) = ( z -1 ) / 3`

`(x - 5)/(5 lambda + 2 ) = ( y -2)/-5 = (z - 1)/ 1 ; x/1 = ( y + 1/2)/(2 lambda ) = ( z -1 ) / 3`

Direction vectors of line are

`(5 lambda + 2 ) hat (i) - 5 hat (j) + hat ( k) " and hat (i) + 2 lambda hat(j)+ 3 hat ( k) `

Lines are perpendicular

∴ there dot product = 0

⇒ ( 5λ + 2) .1-5.(2λ ) + 1.3 = 0

5λ + 2 - 10λ +3 = 0

-5λ + 5 = 0

-5λ =- 5

`λ = (-5)/-5`

λ = 1

Put λ = 1

`(x - 5 ) / 7 = ( y -2 )/-5 = (z - 1) /1 = t `

⇒ x = 7t + 5 , y = -5t + 2 , z = t +1

`x/1 = (y + 1/2 )/2 = (z-1)/3 = s`

x = s ; y =2s -`1/2` , z = 3s + 1

It lines are intersecting their x, y and z coordinate will be same equaiting x

⇒ 7t + 5 = s

s - 7t = 5 .......(i)

Equating z

t + 1 = 3s + 1

t = 3s .......... (ii)

s - 21s = 5

`s = - 1/4 t = -3/4`

Now for first line

`y = -5t + 2 = - 15/4 + 2 = - 23/4`

For second line

`y = 2s -1/5 = 2 xx (-1)/4 - 1/2 = -1`

y co-ordinates are not equal

so the lines are not intersecting.