The Planes

The general form ax + by

The general form ax + by + cz + d = 0 represents any plane in 3D space. Here, (a, b, c) gives the direction of the normal vector to the plane. To solv

For plane through point (x₁

For plane through point (x₁, y₁, z₁): a(x - x₁) + b(y - y₁) + c(z - z₁) = 0. For three points, use determinant method: |x-x₁ y-y₁ z-z₁|/|x₂-x₁ y₂-y₁ z

When plane cuts axes at (a

When plane cuts axes at (a,0,0), (0,b,0), (0,0,c), equation becomes x/a + y/b + z/c = 1. To convert general form to intercept form: Step 1 - Rearrange

Equation lx + my + nz

Equation lx + my + nz = p where l, m, n are direction cosines of normal and p is perpendicular distance from origin. To convert ax + by + cz + d = 0 t

For planes a₁x + b₁y +

For planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0, angle θ satisfies: cos θ = |a₁a₂ + b₁b₂ + c₁c₂|/[√(a₁² + b₁² + c₁²) × √(a₂² + b₂² +