Shortest distance between the lines (x-1)/2 = (y+8)/-7 = (z-4)/5 and (x-1)/2 = (y-2)/1 = (z-6)/-3 is

Q: Why can't I just calculate the distance between the points $(1, -8, 4)$ and $(1, 2, 6)$?

That is a classic trap! Those points are just arbitrary starting points on the lines. The shortest distance is not the distance between these specific points, but the distance between the lines themselves. The shortest path is a unique segment perpendicular to both lines, which likely connects two entirely different points on the lines.

Q: Does the order of the cross product $\vec{b}_1 \times \vec{b}_2$ matter?

Mathematically, $\vec{b}_1 \times \vec{b}_2 = -(\vec{b}_2 \times \vec{b}_1)$. However, because we take the *magnitude* in the denominator and the *absolute value* in the numerator, the sign cancels out. You will get the same distance regardless of the order, so don't stress about it!

Q: What if the lines were parallel?

If the lines were parallel, their direction vectors $\vec{b}_1$ and $\vec{b}_2$ would be proportional. In that case, the cross product would be the zero vector, and the formula would fail. For parallel lines, you would simply find the distance from a point on one line to the other line.

Question

Shortest distance between the lines $\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$ and $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$ is

Options:

(A) $2\sqrt{3}$

(B) $4\sqrt{3}$

(C) $3\sqrt{3}$

(D) $5\sqrt{3}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

In three-dimensional space, the shortest distance between two skew lines is the length of the unique line segment that is simultaneously perpendicular to both lines. This segment acts as the bridge connecting the two lines at their closest points.

To calculate this distance, we utilize the following formula:

$$d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 	imes \vec{b}_2)|}{|\vec{b}_1 	imes \vec{b}_2|}$$

This formula represents the projection of the vector connecting any two points on the lines onto the direction of their common perpendicular.

### Extracting Line Parameters

We begin by identifying the points and direction vectors from the symmetric equations of the lines.

For the first line, $\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$, we extract:
*   Point $\vec{a}_1 = (1, -8, 4)$
*   Direction vector $\vec{b}_1 = 2\hat{i} - 7\hat{j} + 5\hat{k}$

For the second line, $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$, we extract:
*   Point $\vec{a}_2 = (1, 2, 6)$
*   Direction vector $\vec{b}_2 = 2\hat{i} + \hat{j} - 3\hat{k}$

### Calculating the Connecting Vector

Next, we determine the vector connecting the two points $\vec{a}_1$ and $\vec{a}_2$:

$$\vec{a}_2 - \vec{a}_1 = (1-1)\hat{i} + (2 - (-8))\hat{j} + (6-4)\hat{k} = 0\hat{i} + 10\hat{j} + 2\hat{k}$$

### Finding the Common Perpendicular

We find the common perpendicular direction $\vec{n}$ by calculating the cross product of the two direction vectors:

$$\vec{n} = \vec{b}_1 	imes \vec{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & -7 & 5 \ 2 & 1 & -3 \end{vmatrix}$$

Expanding the determinant, we obtain:

$$\vec{n} = \hat{i}(21 - 5) - \hat{j}(-6 - 10) + \hat{k}(2 - (-14)) = 16\hat{i} + 16\hat{j} + 16\hat{k}$$

Factoring out the constant, we have $\vec{n} = 16(\hat{i} + \hat{j} + \hat{k})$. The magnitude of this vector is:

$$|\vec{n}| = 16\sqrt{1^2 + 1^2 + 1^2} = 16\sqrt{3}$$

### Final Calculation

Now, we compute the dot product of the connecting vector and the normal vector:

$$(\vec{a}_2 - \vec{a}_1) \cdot \vec{n} = (0\hat{i} + 10\hat{j} + 2\hat{k}) \cdot (16\hat{i} + 16\hat{j} + 16\hat{k}) = 0 + 160 + 32 = 192$$

Substituting these values into our distance formula:

$$d = \frac{|192|}{16\sqrt{3}} = \frac{12}{\sqrt{3}}$$

Rationalizing the denominator, we arrive at the final result:

**$d = 4\sqrt{3}$**

Animated Solution for Mathematics - Three Dimensional Geometry: Shortest distance between the lines $\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}$ and $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}$ is

Select Answer:

Visualized Solution

Visualizing Skew Lines

Parameters of Line 1

Parameters of Line 2

Vector Connecting the Lines

Common Perpendicular Direction

Evaluating the Determinant

Magnitude of Normal Vector

Numerator: The Dot Product

Calculating Shortest Distance

The Sigma Insight: Shortest Distance Between Two Skew Lines

Analyzing the Setup

Extracting Line Parameters

Calculating the Connecting Vector

Finding the Common Perpendicular

Final Calculation

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