LINES AND PLANES IN SPACE

Equation of a line in space. Let be the position vector of a point P in space. Then the vector equation of a line passing through point P in a direction given by vector is

where t is a parameter that ranges over the real numbers. If

then 1) is equivalent to the following parametric representation of a straight line:

As the parameter t varies over the real numbers the vector generates the line.

Equation of a plane in space. Let be the position vector of a point P in space. Then the vector equation of a plane passing through point P and parallel to two noncollinear vectors and is

where parameters u and v range independently over the real numbers. If

then 3) is equivalent to the following parametric representation of a plane:

As the parameters u and v vary over the real numbers the vector generates the plane.

Normal and one point form of the equation of a plane. Let be the position vector of a point P in space. Then the vector equation of a plane passing through point P and perpendicular to some vector (i.e. a normal to the plane) is

 If

5) is equivalent to

6)        (x₁ - a₁)n₁ + (x₂ - a₂)n₂ + (x₃ - a₃)n₃ = 0

Three point form of the equation of a plane. Let be the position vectors of three noncollinear points P₁, P₂ and P₃ in space. Then the vector equation of the plane passing through points P₁, P₂ and P₃ is

Proof. The vectors and represent two linearly independent vectors in the plane and so represents a vector normal to the plane. Equation 7) then follows from equation 5) above.

If

7) is equivalent to

Reference.

  Lipschutz. Differential Geometry. Chap. 2