Unit 11.5: Lines and Planes

Vector equation of a line

r=r₀+tv

Parametric line equations

x=x₀+at, y=y₀+bt, z=z₀+ct

Symmetric line form

(x-x₀)/a=(y-y₀)/b=(z-z₀)/c

Plane equation

a(x-x₀)+b(y-y₀)+c(z-z₀)=0

General plane form

ax+by+cz=d

Normal vector of plane

⟨a,b,c⟩

Distance from point to plane

|ax₀+by₀+cz₀-d|/√(a²+b²+c²)

Angle between planes

Angle between their normal vectors.

Angle between line and plane

90° − angle(line direction, plane normal)

How to find a plane from three points

Take two direction vectors and cross them.

How to find line intersection

Set parametric equations equal and solve.

How to tell if lines are parallel

Direction vectors are multiples.

How to tell if lines intersect

Solve for common point.

How to tell if lines are skew

Not parallel and do not intersect.

Professor trick

If equation is ax+by+cz=d, the coefficients immediately give the normal vector.