Q.Find the shortest distance between the lines $\frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}$ and $\frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}$

Concept: Shortest Distance Between Skew Lines — the distance is the length of the common perpendicular between two non-parallel, non-intersecting lines.

Step 1: Identify direction vectors and points.

Line 1: point $A(-1,-1,-1)$, direction $\vec{b_1} = (7, -6, 1)$.

Line 2: point $B(3,5,7)$, direction $\vec{b_2} = (1, -2, 1)$.

Step 2: Check if lines are parallel.

$\frac{7}{1} \neq \frac{-6}{-2} \neq \frac{1}{1}$, so not parallel — they are skew.

Step 3: Use the formula for shortest distance $d$:

where $\vec{AB} = B - A = (4, 6, 8)$.

Step 4: Compute cross product and dot product. …

Concept & Intuition

The two lines given are not parallel (their direction ratios are not proportional), and they are not intersecting (we can check quickly). They are skew lines — they lie in parallel planes but never meet. The shortest distance between them is the length of the common perpendicular segment that connects a point on one line to a point on the other.

Think of it this way: imagine two straight wires floating in space, not parallel and not touching. The shortest distance between them is the length of the line segment that is perpendicular to both wires simultaneously. That perpendicular direction is given by the cross product of the two direction vectors.

The formula we use is a compact way of saying: take any point on line 1 and any point on line 2, form the vector joining them, then project that vector onto the unit vector perpendicular to both lines. The magnitude of that projection is the shortest distance.

Step-by-Step Solution

1. Identify the vectors from the given equations.

The first line is $\frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}$. This is in symmetric form. A point on this line is obtained by setting each numerator to zero: $(-1, -1, -1)$. The direction ratios are the denominators: $(7, -6, 1)$.

So for line 1:

The second line is $\frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}$. A point on it is $(3, 5, 7)$, and direction ratios are $(1, -2, 1)$.

So for line 2:

2. Find the vector joining the two points.

3. Compute the cross product of the direction vectors.

The direction perpendicular to both lines is $\vec{b}_1 \times \vec{b}_2$.

Expanding:

Method: Shortest distance between skew lines given in Cartesian form

The shortest distance between two non-parallel, non-intersecting lines is the length of their common perpendicular. When the lines are in symmetric form, first read a point and a direction from each, then apply the standard formula.

Steps

Step 1: Read point and direction from each line. For $\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$, the point is $(x_1,y_1,z_1)$ (numerators) and the direction is $(a,b,c)$ (denominators). Do this for both lines to get $\vec a_1,\vec b_1$ and $\vec a_2,\vec b_2$.

Step 2: Cross the directions. Compute $\vec b_1\times\vec b_2$ and its magnitude. …

Common Mistakes

Mistake 1: Assuming the lines are parallel or intersecting without checking.

Why it's wrong: the direction ratios $(7,-6,1)$ and $(1,-2,1)$ aren't proportional, so the lines are skew and the common-perpendicular formula applies. Correct approach: confirm non-proportional directions, then use the skew-distance formula.

Mistake 2: Sign errors reading points from the symmetric form. …