Q.If the lines $\dfrac{x-1}{-3} = \dfrac{y-2}{2\lambda} = \dfrac{z-3}{2}$ and $\dfrac{x-1}{3\lambda} = \dfrac{y-1}{2} = \dfrac{z-6}{-5}$ are perpendicular, find the value of $\lambda$. Hence find whether the lines are intersecting or not.

Concept: Two lines are perpendicular when the dot product of their direction vectors is zero.

Step 1 – Direction vectors

From the first line: $\vec{d_1} = (-3,\; 2\lambda,\; 2)$

From the second line: $\vec{d_2} = (3\lambda,\; 2,\; -5)$

Step 2 – Perpendicular condition

$\vec{d_1} \cdot \vec{d_2} = 0$

$(-3)(3\lambda) + (2\lambda)(2) + (2)(-5) = 0$

$-9\lambda + 4\lambda - 10 = 0 \quad\Rightarrow\quad -5\lambda = 10 \quad\Rightarrow\quad \lambda = -2$

Step 3 – Check intersection

For $\lambda = -2$, the lines become:

Line 1: $\frac{x-1}{-3} = \frac{y-2}{-4} = \frac{z-3}{2}$

Line 2: $\frac{x-1}{-6} = \frac{y-1}{2} = \frac{z-6}{-5}$

Point on line 1: $(1,2,3)$; point on line 2: $(1,1,6)$. …

Concept and Intuition

When two lines in 3D are given in symmetric form, their direction vectors are read directly from the denominators. For perpendicularity, the dot product of these direction vectors must be zero. That gives us an equation to solve for $\lambda$.

Once we have $\lambda$, the question of whether the lines intersect is separate. Two lines in 3D can be parallel, intersecting, or skew (non-parallel and non-intersecting). Since we already know they are perpendicular (hence not parallel), we only need to check if they share a common point. If not, they are skew.

Step-by-step Solution

1. Extract direction vectors from the symmetric equations.

For the first line:

The direction vector is:

For the second line:

The direction vector is:

2. Apply the perpendicularity condition.

Two vectors are perpendicular if their dot product is zero:

Compute:

3. Write the lines with $\lambda = -2$.

First line:

Direction: $\vec{d_1} = (-3, -4, 2)$

Point on line: $A(1, 2, 3)$

Second line:

Direction: $\vec{d_2} = (-6, 2, -5)$

Point on line: $B(1, 1, 6)$

4. Check if the lines intersect.

Two lines intersect if there exist parameters $t$ and $s$ such that:

Write the parametric equations:

From first line:

From second line: