Before You Begin

Question #40745

Difficulty: 8/10 | 🟢 Free

The Question

Consider two lines $L_1$ and $L_2$ in $\mathbb{R}^3$ . Line $L_1$ passes through the point with position vector $\vec{a}_1 = \hat{i} + 2\hat{j} - \hat{k}$ and is parallel to the vector $\vec{d}_1 = \vec{u} \times (\vec{v} \times \vec{w})$ , where $\vec{u} = \hat{i} + 2\hat{j} - \hat{k}$ , $\vec{v} = 2\hat{i} - \hat{j} + 3\hat{k}$ , and $\vec{w} = \hat{i} - \hat{j} + \hat{k}$ . Line $L_2$ passes through the point with position vector $\vec{a}_2 = 2\hat{i} - \hat{j} + \hat{k}$ and is parallel to the vector $\vec{d}_2 = \vec{x} \times (\vec{y} \times \vec{z})$ , where $\vec{x} = \hat{i} - \hat{j}$ , $\vec{y} = \hat{i} + \hat{j} + 3\hat{k}$ , and $\vec{z} = \hat{i} + \hat{k}$ . The shortest distance between $L_1$ and $L_2$ is:

Options

A.
$\sqrt{\frac{138}{11}}$
B.
$0$
C.
$\sqrt{\frac{135}{11}}$
D.
$\sqrt{11}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Distance Between Parallel Lines|Geometric Interpretation of Cross Product|Vector Triple Product Expansion

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly identify the geometric relationship between two lines before applying distance formulas.

By obfuscating the direction vectors using vector triple products, students are forced to evaluate them algebraically.

The lethal trap lies in the fact that the evaluated direction vectors are parallel ( $\vec{d}_1 = -\vec{d}_2$ ).

A student who blindly applies the standard shortest distance formula for skew lines, $d = |[\vec{a}_2 - \vec{a}_1, \vec{d}_1, \vec{d}_2]| / |\vec{d}_1 \times \vec{d}_2|$ , will find the numerator (scalar triple product) evaluates to $0$ .

A common, fatal mistake is to immediately conclude the lines intersect and the distance is $0$ , failing to realize the denominator is also $0$ (creating a $0/0$ indeterminate form).

Recognizing the lines are parallel requires shifting to the parallel lines distance formula, rewarding conceptual clarity over rote memorization.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Distance Between Parallel Lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free