Question

Check whether the lines given by \(\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4}\) and \(\dfrac{x-4}{5} = \dfrac{y-1}{2} = z\) are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

Hint:

The direction ratios 2,3,4 and 5,2,1 are not proportional. Equate general points of both lines to find lambda = mu = -1 and the point (-1,-1,-1).

Answer 1

Step 1: Read the direction ratios:
The first line passes through \((1, 2, 3)\) with direction ratios \(2, 3, 4\).
The second line passes through \((4, 1, 0)\) with direction ratios \(5, 2, 1\).
The last fraction \(z\) is the same as \(\dfrac{z-0}{1}\).

Step 2: Test for parallel lines:
Lines are parallel only if the direction ratios are proportional.
\[ \frac{2}{5} \neq \frac{3}{2} \neq \frac{4}{1} \]
The ratios are not equal, so the lines are not parallel.

Step 3: Write general points on both lines:
Let each ratio of the first line equal \(\lambda\) and each ratio of the second line equal \(\mu\).
\[ (2\lambda + 1,\; 3\lambda + 2,\; 4\lambda + 3) \quad \text{and} \quad (5\mu + 4,\; 2\mu + 1,\; \mu) \]

Step 4: Equate the coordinates:
If the lines meet, a common point exists. We get three equations.
\[ 2\lambda + 1 = 5\mu + 4, \quad 3\lambda + 2 = 2\mu + 1, \quad 4\lambda + 3 = \mu \]
The first two simplify to \(2\lambda - 5\mu = 3\) and \(3\lambda - 2\mu = -1\).
From the first, \(\lambda = \dfrac{3 + 5\mu}{2}\). Put this in the second.
\[ \frac{3(3+5\mu)}{2} - 2\mu = -1 \;\Rightarrow\; 9 + 15\mu - 4\mu = -2 \;\Rightarrow\; \mu = -1 \]
Then \(\lambda = \dfrac{3 - 5}{2} = -1\).

Step 5: Check the third equation:
Left side: \(4(-1) + 3 = -1\). Right side: \(\mu = -1\). They are equal, so the lines intersect.

Step 6: Find the point of intersection:
Put \(\lambda = -1\) in the first line.
\[ (2(-1) + 1,\; 3(-1) + 2,\; 4(-1) + 3) = (-1, -1, -1) \]
The second line at \(\mu = -1\) gives \((-1, -1, -1)\) too.

Final Answer:
The lines are not parallel. They intersect at the point below. \[ \boxed{(-1, -1, -1)} \]