Distance Between Two Lines

First, convert the equations into vector form. The first line can be written as:

Line 1: .

The second line can be written as:

Line 2: .

The shortest distance between two skew lines is along the common perpendicular. The direction vector of the perpendicular is given by the cross product of the direction vectors of the two lines:

, .

Cross product is:

.

If and , the vector between two points on each line is .

The shortest distance is found by projecting onto :

.

First calculate the dot product:

.

The magnitude of the cross product:

.

Thus, the distance is:

This matches the correct answer option: .

To find the shortest distance between two skew lines, we use the formula:

where and are points on the lines, and and are direction vectors.

For the first line:

For the second line:

Calculate :

Find the cross product :

Calculate the magnitude:

Now, find the dot product :

The shortest distance is: