Line and Plane
Easy Overview
Remember line and plane? It's back, but this time in 3D. We're figuring out where lines and planes intersect, the distance between them, and how to find angles between a line and a plane. Think of it as the geometry of where things meet — like figuring out where a laser beam hits a wall.
Equation of a Plane
A plane is a flat surface that extends forever. Vector form: r·n = d, where n is the normal (perpendicular) vector. Cartesian: ax + by + cz = d. If d = 0, the plane passes through the origin. The coefficients (a, b, c) tell you which direction the plane faces — they're the normal vector.
Plane Through Three Points
Three non-collinear points define a unique plane. Find two vectors in the plane (subtract points), then cross product gives the normal. Then use r·n = d with any point. It's like setting up a tent — three poles determine the flat surface.
Angle Between Two Planes
The angle between two planes is the same as the angle between their normals. Two normals n₁ and n₂: cos θ = |n₁·n₂| / (|n₁||n₂|). If n₁·n₂ = 0, the planes are perpendicular. If normals are parallel, the planes are parallel. Simple as that.
Angle Between a Line and a Plane
A line meets a plane at some angle. Find the complement of the angle between the line and the normal. If line direction is b and plane normal is n: sin φ = |b·n| / (|b||n|). Why sine? Because you're measuring how much the line is NOT parallel to the plane.
Distance of a Point from a Plane
How far is a point from a plane? Drop a perpendicular — distance = |ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²). The numerator is plugging the point into the plane equation, absolute value. Denominator is the normal's length. It's like measuring how high a balloon is above a flat roof.
Line of Intersection of Two Planes
Two non-parallel planes meet along a straight line. The direction of that line is perpendicular to BOTH normals — so take n₁ × n₂. To find a point on the line, set one coordinate to zero and solve the plane equations. You now have a point and a direction — that's the line.
Key Points
- •Plane: ax + by + cz = d, where (a,b,c) is the normal vector
- •Three points → two vectors → cross product → normal → plane equation
- •Angle between planes = angle between normals: cos θ = |n₁·n₂| / (|n₁||n₂|)
- •Angle between line and plane: sin φ = |b·n| / (|b||n|)
- •Distance from point (x₁,y₁,z₁) to plane ax+by+cz=d: |ax₁+by₁+cz₁−d| / √(a²+b²+c²)
- •Intersection of two planes: direction = n₁ × n₂
Practice Questions
- Find the equation of the plane passing through (1, 2, 1) with normal perpendicular to (2, 1, 0) and (3, 1, −1).
- Find the distance of point (2, 3, 1) from the plane 2x + 3y − z + 5 = 0.
- Find the angle between the line (x−1)/2 = y/3 = (z+2)/4 and the plane 3x + 4y − 5z = 7.
- Find the equation of the line of intersection of planes x + y + z = 1 and 2x − y + 3z = 4.