Planes in 3D: Equations, Distance, Angle & Foot of Perpendicular

• Write the equation of a plane in vector form, Cartesian form, and intercept form
• Find the distance from a point to a plane
• Find the angle between two planes (dihedral angle)
• Determine if two planes are parallel or perpendicular
• Find the equation of a plane through three given points
• Find the foot of the perpendicular from a point to a plane

Level 1 — Foundations

A plane is determined by a point on it and its normal vector (perpendicular to the plane).

Vector form: r·n = d, where n is the normal vector and d is a constant.

If n̂ is a unit normal, then r·n̂ = D, where D is the perpendicular distance from origin to the plane.

Cartesian form: Ax + By + Cz + D = 0, where (A, B, C) is the normal vector.

Intercept form (plane cutting x-axis at a, y-axis at b, z-axis at c): $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

Normal vector to Ax + By + Cz + D = 0 is the vector (A, B, C).

Level 2 — JEE depth

Distance from point (x₀, y₀, z₀) to plane Ax+By+Cz+D=0: $d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$

Angle between two planes Ax+By+Cz+D₁=0 and A'x+B'y+C'z+D₂=0: $\cos\theta = \frac{|AA' + BB' + CC'|}{\sqrt{A^2+B^2+C^2}\cdot\sqrt{A'^2+B'^2+C'^2}}$

Perpendicular planes: $AA' + BB' + CC' = 0$

Parallel planes: $\dfrac{A}{A'} = \dfrac{B}{B'} = \dfrac{C}{C'}$ (normals are parallel)

Plane through three points P₁(x₁,y₁,z₁), P₂(x₂,y₂,z₂), P₃(x₃,y₃,z₃): $\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$

Foot of perpendicular from P(x₀,y₀,z₀) to plane Ax+By+Cz+D=0:

The foot Q is given by: $\frac{x_Q - x_0}{A} = \frac{y_Q - y_0}{B} = \frac{z_Q - z_0}{C} = -\frac{Ax_0+By_0+Cz_0+D}{A^2+B^2+C^2}$

Image (reflection) of point in the plane: P' = 2Q − P

Family of planes through the intersection of two planes P₁=0 and P₂=0: $P_1 + \lambda P_2 = 0 \quad (\lambda \in \mathbb{R})$

Find the distance from the point P(3, −1, 2) to the plane 2x + y − 2z + 3 = 0.
Also find the foot of the perpendicular from P to the plane.

Step 1: Identify A, B, C, D
  A=2, B=1, C=−2, D=3

Step 2: Distance
  d = |2(3) + 1(−1) + (−2)(2) + 3| / √(4+1+4)
  d = |6 − 1 − 4 + 3| / √9
  d = |4| / 3 = 4/3

Step 3: Foot of perpendicular
  k = −(Ax₀+By₀+Cz₀+D)/(A²+B²+C²)
  k = −(2(3)+1(−1)+(−2)(2)+3)/(4+1+4)
  k = −(6−1−4+3)/9 = −4/9

  x_Q = x₀ + Ak = 3 + 2(−4/9) = 3 − 8/9 = 19/9
  y_Q = y₀ + Bk = −1 + 1(−4/9) = −1 − 4/9 = −13/9
  z_Q = z₀ + Ck = 2 + (−2)(−4/9) = 2 + 8/9 = 26/9

  Foot Q = (19/9, −13/9, 26/9)

Verify: distance |PQ| = √((19/9−3)² + (−13/9+1)² + (26/9−2)²)
  = √((−8/9)² + (−4/9)² + (8/9)²)
  = √(64+16+64)/9 = √144/9 = 12/9 = 4/3 ✓

• Find the equation of a plane passing through a given point with a given normal vector
• Find the equation of a plane through three given points
• Find the distance from the origin to a given plane
• Find the angle between two planes and state whether they are perpendicular or parallel
• Find the equation of the plane parallel to a given plane and at a given distance from it

• Fig 11.12: Normal vector n to a plane and position vector r of a general point
• Fig 11.14: Three points determining a unique plane
• Fig 11.17: Foot of perpendicular from an external point to a plane

• What is the normal vector of the plane 3x − 2y + 5z = 7? → (3, −2, 5)
• Are the planes 2x+3y−z=4 and 4x+6y−2z=9 parallel? → Yes (ratios 2:4=3:6=−1:−2)
• Distance from origin to plane x+y+z=3: → 3/√3 = √3
• If two planes are perpendicular, their normals satisfy ___? → n₁·n₂ = 0
• Stretch: Find the equation of the plane that bisects perpendicularly the line segment joining (1,2,3) and (3,4,5).

NCERT Chapter 11 link: Three Dimensional Geometry — planes covered in sections 11.5–11.8 with Exercises 11.3 and Miscellaneous
Exam connections: Line-plane intersection, distance problems, and family of planes appear in every JEE Main; angle between line and plane uses the complement of angle between line and normal
Study strategy: Solve all NCERT Ex 11.3 problems first; then practice 5 JEE Main problems on foot of perpendicular and family of planes — these are the most common question types.

• 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.