Question

Find whether the lines r= (i-j - k) +lambda (2i + j) and r = (2i-j) + u(î +j - k) intersect or not.If intersecting, find their point of intersection.​

Answer 1

Therefore the lines intersect each other at (3,0,1).

Step-by-step explanation:

Given lines are

$\vec{r}=\hat i-\hat j-\hat k +\lambda (2\hat i +\hat j)$

$\Rightarrow \vec{r} = (1+2\lambda)\hat i+(-1+\lambda)\hat j-\hat k$ .....(1)

and

$\vec{r}= 2 \hat i-\hat j +\mu(\hat i+\hat j-\hat k)$

$\Rightarrow \vec{r}=(2+\mu)\hat i+(-1+\mu)\hat j- \mu \hat k$ ....(2)

For the intersection

$(1+2\lambda)\hat i+(-1+\lambda)\hat j-\hat k=(2+\mu)\hat i+(-1+\mu)\hat j- \mu \hat k$

Therefore

$1+2\lambda =2+\mu$   ,       $-1+\lambda=-1+\mu$   and  $-1=-\mu$

From the third equation we get

$\mu =1$

Putting the value of μ in the first equation

$1+2\lambda =2+1$

$\Rightarrow \lambda =1$

we get a definite value of  μ  and λ.

Putting the value of μ in second line

$\vec r =(2+1)\hat i +(-1+1)\hat j-1 \hat k$

  $=3\hat i -\hat k$

Therefore the lines intersect each other at (3,0,1).