Calculus 3 Lecture Notes: Equations of Lines and Planes

Equations of Lines in 3D Space

• Key Concepts

  • A line in space is determined by a point and a direction.
  • The direction in 3D space is given by a direction vector ( \mathbf{V} ) with components ( (a, b, c) ).

• Vector Form of a Line

  • Given a point ( P_0(x_0, y_0, z_0) ) and a direction vector ( \mathbf{V} ), the line is parallel to ( \mathbf{V} ).
  • The vector from ( P_0 ) to any point ( P(x, y, z) ) on the line is a scalar multiple of ( \mathbf{V} ).
  • Vector equation: ( \mathbf{R} = \mathbf{R}_0 + t \mathbf{V} )

• Parametric Equations

  • ( x = x_0 + ta )
  • ( y = y_0 + tb )
  • ( z = z_0 + tc )

• Symmetric Equations

  • Derived by eliminating the parameter ( t ).
  • ( \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} )
  • If one of ( a, b, c ) is zero, the corresponding coordinate is constant.

Examples and Practice

• Find parametric equations for a line through a given point and direction.
• Symmetric Equations Example: If direction vector ( V = (5, -2, 0) ), then ( z = 3 ) is constant.

Determining Parallel and Perpendicular Lines

• Parallel Lines: Direction vectors are scalar multiples.
• Perpendicular Lines: Dot product of direction vectors is zero.
• Skew Lines: Non-parallel lines in 3D that do not intersect.

Distance from a Point to a Line

• Use the cross product to find the shortest distance.
• Method: Verify with Pythagorean theorem, vector projections, or trig properties.

Equations of Planes

• Plane Determination

  • A plane is defined by a point and a direction, specifically a normal vector ( \mathbf{N} ).

• Vector Form of Plane

  • Dot product of ( \mathbf{N} ) and any vector in the plane is zero.

• Scalar Equation of Plane

  • Given normal vector ( \mathbf{N} = (A, B, C) ) and point ( P_0 ):
  • Equation: ( A(x-x_0) + B(y-y_0) + C(z-z_0) = 0 )
  • Simplifies to ( Ax + By + Cz = D ) (linear standard form).

Graphing Planes

• Normal Vector: Read off coefficients.
• Intersections and Traces: Set variables to zero, graph traces in planes.

Finding Equation of Planes

• Three Points Determine a Plane
  • Use cross product of vectors between points to find normal vector.
• Plane Containing a Line
  • Use direction vector of line and another vector in the plane.

Intersection of Planes

• Line of Intersection
  • Find point on both planes and direction vector by cross product of normals.

Distance from Point to Plane

• Method: Use vector projections.

• Equation: ( D = \frac{|\mathbf{PQ} \cdot \mathbf{N}|}{|\mathbf{N}|} ), ( \mathbf{PQ} ) is vector from point to plane.

• Memorization of textbook formulas is discouraged; understanding and application are preferred.