Equation of a Plane in 3D Geometry (JEE Advanced)

“equation of a plane, distance of a point from a plane, angle between two planes, angle between a line and the plane, coplanar lines”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: Equation of a line; Skew lines, the shortest distance between them and its equation.

JEE Main focuses on lines in 3D and skew lines, but Advanced adds planes, which means you need to handle plane equations, point-to-plane distances, and angles involving planes. These concepts appear frequently in Advanced and require comfort with vector operations you may not have practiced much.

• A plane with normal vector (a, b, c) passing through point (x₀, y₀, z₀) has equation a(x − x₀) + b(y − y₀) + c(z − z₀) = 0
• The general form ax + by + cz + d = 0 tells you the normal vector is (a, b, c)
• Three non-collinear points determine a unique plane. Find two vectors in the plane, take their cross product to get the normal
• The angle between two planes equals the angle between their normal vectors
• A line is parallel to a plane if the direction vector of the line is perpendicular to the normal of the plane
• Two lines are coplanar if they either intersect or are parallel. Use the scalar triple product to test this.

• General equation of plane: ax + by + cz + d = 0
• Vector equation: r⃗ · n⃗ = d, where n⃗ is normal vector
• Distance from point (x₁, y₁, z₁) to plane ax + by + cz + d = 0: |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)
• Angle between two planes with normals n⃗₁ and n⃗₂: cos θ = |n⃗₁ · n⃗₂| / (|n⃗₁| |n⃗₂|)
• Angle between line (direction d⃗) and plane (normal n⃗): sin θ = |d⃗ · n⃗| / (|d⃗| |n⃗|)
• Plane through three points A, B, C: use n⃗ = AB⃗ × AC⃗ as normal
• Coplanarity test for lines: if lines pass through points a⃗, b⃗ with directions p⃗, q⃗, then [b⃗ − a⃗, p⃗, q⃗] = 0 for coplanar lines
• Equation of plane through intersection of planes P₁ and P₂: P₁ + λP₂ = 0

• Forgetting to take absolute value in the distance formula, which can give negative results
• Confusing the angle formula for two planes (uses cosine) with the angle formula for a line and plane (uses sine)
• Using the direction vector of a line as if it were the normal vector of a plane containing that line
• In coplanarity problems, mixing up which vectors go into the scalar triple product

Question. Find the distance from the point (1, 2, 3) to the plane passing through the points A(1, 0, 0), B(0, 2, 0), and C(0, 0, 3).

Solved twice independently; both solutions agree.