Lines in 3D: Direction Cosines, Vector & Cartesian Forms, Skew Lines

• Express a line using direction cosines and direction ratios
• Write the equation of a line in vector form and Cartesian form
• Determine whether two lines are parallel, perpendicular, or skew
• Find the shortest distance between two skew lines
• Find the distance between two parallel lines

Level 1 — Foundations

A line in 3D is fully determined by a point on it and its direction.

Direction cosines (l, m, n): The cosines of the angles α, β, γ that the line makes with the positive x, y, z axes. $l = \cos\alpha,\quad m = \cos\beta,\quad n = \cos\gamma, \quad l^2+m^2+n^2 = 1$

Direction ratios (a, b, c): Any set of numbers proportional to the direction cosines. $l = \frac{a}{\sqrt{a^2+b^2+c^2}},\quad m = \frac{b}{\sqrt{a^2+b^2+c^2}},\quad n = \frac{c}{\sqrt{a^2+b^2+c^2}}$

Vector form of a line through point a with direction b: $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}, \quad \lambda \in \mathbb{R}$

Cartesian form through (x₁, y₁, z₁) with direction ratios (l, m, n): $\frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}$

Level 2 — JEE depth

Angle between two lines with direction cosines (l₁,m₁,n₁) and (l₂,m₂,n₂): $\cos\theta = |l_1l_2 + m_1m_2 + n_1n_2|$

Using direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂): $\cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot\sqrt{a_2^2+b_2^2+c_2^2}}$

Perpendicular lines: $l_1l_2 + m_1m_2 + n_1n_2 = 0$, i.e., $a_1a_2+b_1b_2+c_1c_2=0$

Parallel lines: $\dfrac{l_1}{l_2} = \dfrac{m_1}{m_2} = \dfrac{n_1}{n_2}$, i.e., $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$

Skew lines: Lines that are neither parallel nor intersecting (they lie in different planes). Only possible in 3D.

Shortest distance between skew lines $\mathbf{r} = \mathbf{a_1} + \lambda\mathbf{b_1}$ and $\mathbf{r} = \mathbf{a_2} + \mu\mathbf{b_2}$: $d = \frac{|(\mathbf{a_2} - \mathbf{a_1}) \cdot (\mathbf{b_1} \times \mathbf{b_2})|}{|\mathbf{b_1} \times \mathbf{b_2}|}$

Distance between parallel lines ($\mathbf{b_1} \parallel \mathbf{b_2}$, both equal b): $d = \frac{|(\mathbf{a_2} - \mathbf{a_1}) \times \mathbf{b}|}{|\mathbf{b}|}$

Line through two points A(a) and B(b): $\mathbf{r} = \mathbf{a} + \lambda(\mathbf{b} - \mathbf{a})$

Find the shortest distance between the skew lines:
L1: r = (i + 2j + k) + λ(i − j + k)
L2: r = (2i − j − k) + μ(2i + j + 2k)

Step 1: Identify vectors
  a1 = i + 2j + k,   b1 = i − j + k
  a2 = 2i − j − k,   b2 = 2i + j + 2k

Step 2: a2 − a1 = (2−1)i + (−1−2)j + (−1−1)k = i − 3j − 2k

Step 3: b1 × b2
  | i   j   k  |
  | 1  −1   1  |
  | 2   1   2  |
  = i[(−1)(2)−(1)(1)] − j[(1)(2)−(1)(2)] + k[(1)(1)−(−1)(2)]
  = i[−2−1] − j[2−2] + k[1+2]
  = −3i + 0j + 3k = −3i + 3k

Step 4: |b1 × b2| = √(9 + 0 + 9) = √18 = 3√2

Step 5: (a2−a1)·(b1×b2) = (1)(−3) + (−3)(0) + (−2)(3) = −3 − 0 − 6 = −9

Step 6: d = |−9| / (3√2) = 9/(3√2) = 3/√2 = 3√2/2

• Convert Cartesian line equation to direction ratios and then direction cosines
• Check if two lines are parallel, perpendicular, or skew
• Find the point of intersection of two intersecting lines
• Find the angle between two given lines
• Find shortest distance between two skew lines

• Fig 11.1: Direction angles α, β, γ of a line
• Fig 11.3: Line through a point with given direction vector (vector form)
• Fig 11.8: Two skew lines in 3D with the common perpendicular segment

• If direction ratios are (1, 1, 1), what are the direction cosines? → (1/√3, 1/√3, 1/√3)
• Two lines: l₁/l₂ = m₁/m₂ = n₁/n₂ — what is their relationship? → Parallel
• Can two lines in 2D be skew? → No, only in 3D
• For skew lines, the shortest distance vector is perpendicular to both b₁ and b₂? → Yes
• Stretch: Prove that the line joining (1,2,3) and (3,2,1) is perpendicular to the line joining (−1,1,0) and (0,2,1).

NCERT Chapter 11 link: Three Dimensional Geometry — lines covered in sections 11.2–11.4 with Exercises 11.1 and 11.2
Exam connections: Prerequisite for planes; connects to parametric coordinates and distance formula from Class 11 3D basics (Chapter 12)
Study strategy: Memorise the skew distance formula with the fraction as a whole — numerator is a scalar triple product. Practice 3 skew distance problems to lock it in before moving to planes.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.