Q.Find the distance between the lines $l_1$ and $l_2$ given by $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$ and $\vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k})$.

Concept: Skew Lines — two lines that are neither parallel nor intersecting. Here the direction vectors are identical, so the lines are actually parallel, not skew. The distance between parallel lines is the perpendicular distance from a point on one line to the other line.

Step 1 – Identify direction vector and points

Direction vector: $\vec{d} = 2\hat{i} + 3\hat{j} + 6\hat{k}$

Point on $l_1$: $A(1, 2, -4)$

Point on $l_2$: $B(3, 3, -5)$

Step 2 – Vector between points

$\overrightarrow{AB} = (3-1)\hat{i} + (3-2)\hat{j} + (-5+4)\hat{k} = 2\hat{i} + 1\hat{j} - 1\hat{k}$

Step 3 – Distance formula for parallel lines

Distance $= \frac{|\overrightarrow{AB} \times \vec{d}|}{|\vec{d}|}$

Cross product: …

Why This Approach Works

When two lines are parallel, the shortest distance between them is simply the perpendicular distance from any point on one line to the other line. This is because the lines never meet and maintain a constant separation — like two parallel railway tracks.

The key insight: instead of finding a complicated common perpendicular, we can take any point $A$ on $l_1$ and any point $B$ on $l_2$, then find the component of $\overrightarrow{AB}$ that is perpendicular to the common direction. That perpendicular component is exactly the shortest distance.

Step-by-Step Solution

1. Identify the direction vectors and points

Both lines have the same direction vector:

A point on $l_1$ (taking $\lambda = 0$):

A point on $l_2$ (taking $\mu = 0$):

2. Find the vector joining the two points

3. Compute the cross product $\overrightarrow{AB} \times \vec{d}$

Expanding:

• $\hat{i}$-component: $(1)(6) - (-1)(3) = 6 + 3 = 9$
• $\hat{j}$-component: $-[(2)(6) - (-1)(2)] = -[12 + 2] = -14$
• $\hat{k}$-component: $(2)(3) - (1)(2) = 6 - 2 = 4$

So:

4. Find the magnitude of this cross product …

Method: Distance Between Two Parallel Lines

Use this when two lines share the same direction (their direction vectors are proportional) — the skew-line triple-product formula gives $0/0$, so a different formula is needed.

Steps

Step 1: Confirm the lines are parallel and pick a point on each.

Check that the direction vectors are proportional; call the common direction $\vec{d}$. Take a point $A$ on the first line and $B$ on the second (e.g. the position vectors at $\lambda=0$).

Step 2: Cross the join vector with the direction.

Compute $\overrightarrow{AB}=B-A$, then $\overrightarrow{AB}\times\vec{d}$. Its magnitude measures the part of $\overrightarrow{AB}$ perpendicular to the lines. …

Common Mistakes

Mistake 1: Applying the skew-line triple-product formula to parallel lines.

Why it's wrong: the two lines share the direction $2\hat{i}+3\hat{j}+6\hat{k}$, so $\vec{b_1}\times\vec{b_2}=\vec{0}$ and the skew formula collapses to $\tfrac{0}{0}$. Correct approach: recognise the equal directions and use $d=\tfrac{|\overrightarrow{AB}\times\vec{d}|}{|\vec{d}|}$.

Mistake 2: Not checking the direction vectors first. …