Straight line:

Straight line in 3 dimensional geometry can be defined as intersection of 2 planes. So general equation of straight line can be stated as the equations of both plane together that is general equation of straight line is a₁x + b₁y + c₁z + d₁ = 0, a₂x + b₂y + c₂z + d₂ = 0              ......(1)

So, equation (1) represents straight line which can be obtained by intersection of 2 planes.

Equation of Straight Line in Different Forms:

Symmetrical Form:

• The equation of straight line passing through point P (x₁, y₁, z₁) and the direction cosines of which are l, m, n is 

• The equation of straight line passing through 2 points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) is

      .

Note:

• The general coordinates of point on a line can be given by (x₁ + lr, y₁ + mr, z₁ + nr) where r is distance between point (x₁, y₁, z₁) and point whose coordinates is to be written.

Example. Find out the equations of straight lines through point (a, b, c) which are

(a) parallel to z axis

(b) perpendicular to z axis

Solution:  (i)  The equation of straight lines parallel to z axis have a = 90⁰, b = 90⁰, g = 0⁰

            => l = 0, m = 0, n = 1

     Thus equation of straight line is parallel to z axis and passing through (a, b, c) is  

(ii)  Equation of straight lines perpendicular to z axis

     Let they make a, b angle with x and y axes respectively.

     Then the equation of straight lines perpendicular to z axis and passing through point (a, b, c) is  

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