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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