Distance Between Two Lines

Concept: 3D Geometry - Shortest Distance Between Two Skew Lines.

Step 1: Identify the passing points and direction vectors of both lines. For line , it passes through point and its direction vector is . For line , it passes through point and its direction vector is . Calculate the vector connecting the points on the two lines: .

Step 2: Calculate the scalar triple product for the numerator. The shortest distance is given by . The numerator is the determinant formed by the components of and : Numerator = .
Expanding along the first row: . Simplify: .

Step 3: Calculate the magnitude of the cross product for the denominator. Find the cross product .
.
The magnitude .
Expand the squares: . Combine like terms: .

Step 4: Set up the equation using the given shortest distance. Substitute the numerator and denominator into the distance formula and equate it to the given value : .
To remove the absolute value and the square roots, square both sides of the equation: .
.

Step 5: Solve the quadratic equation for . Cross-multiply the equation: . .
Move all terms to the left side: .
Divide the entire equation by 5 to simplify: .
Factor the quadratic: . The possible values are or . The sum of these possible values is . $ $

Concept: The shortest distance between two skew lines is given by: where are points on the lines and are their direction vectors.
Step 1: {Identify points and direction vectors.} First line: Point Direction vector Second line: Point Direction vector
Step 2: {Find .}
Step 3: {Find vector between points.}
Step 4: {Compute scalar triple product.}
Step 5: {Find magnitude of cross product.}
Step 6: {Compute shortest distance.} After simplification according to given options,