Question

Lines $\bar r = (3 + t)\hat i + (i - t)\hat j + ( - 2 - 2t)\hat k$,$t \in R$ and $x = 4 + k,y = - k,z = - 4 - 2k,$$k \in R$, then relations between lines is
A) Skew
B) Coincident
C) Parallel
D) Perpendicular

Answer 1

Hint: First line is given in vector form and other one is given in Cartesian form. Convert the second line into vector form and check if the lines are the same or if their direction ratios are the same by comparing both the equations.

Complete step by step answer:
Here the first line given is $\bar r = (3 + t)\hat i + (1 - t)\hat j + ( - 2 - 2t)\hat k$.
This can be written as $\bar r = 3\hat i + t\hat i + \hat j - t\hat j - 2\hat k - 2t\hat k$.
Separating the terms having $t$ together, we’ll get:
$ \Rightarrow \bar r = 3\hat i + \hat j - 2\hat k + t(\hat i - \hat j - 2\hat k) - - - - - - - > (1)$
And here for the second line they have given the points as $x = 4 + k,y = - k,z = - 4 - 2k.$
We have to form the line by using the given points.
Let us consider the line is of the form $\bar r = x\hat i + y\hat j + z\hat k$
Now let us substitute the given points in above form to get equation of line as:
$
   \Rightarrow \vec r = (4 + k)\hat i + ( - k)\hat j + ( - 4 - 2k)\hat k \\
   \Rightarrow \vec r = 4\hat i + k\hat i - k\hat j - 4\hat k - 2\hat k \\
$
Separating the terms having $k$ together, we’ll get:
$ \Rightarrow \vec r = 4\hat i - 4\hat k + k(\hat i - \hat j - 2\hat k) - - - - - - - - - - > (2)$
Now on comparing equation (1) and (2) we get to know that the direction ratios of both the lines are the same and it is $(1, - 1, - 2)$.
We know that if two lines have the same direction ratios then the lines are parallel.

Therefore we can say that the given lines are parallel. Option C is the correct answer.

NOTE: In this type of problems we have found the equations by expanding them in proper format that means in the term of x, y, z points. Later we have to find the direction ratio, when it comes to this problem we can say that the direction of given lines are the same .As we know that if the given lines have the same direction then they are parallel.