﻿ Line in 3D space
Line 3D geometry - defined by 2 points
 Line defined by 2 points Point P1(x1,y1,z1) ( , , ) Point P2(x2,y2,z2) ( , , )
Distance from
P1 to P2 Line spherical angles  Parametric
equation of the line
x = + t
y = + t
z = + t
 If   t = The equivalent point on the line is: If   t = The equivalent point on the line is:
Line equation x +
= y +
= z +
 Direction anglesDirection cosines       Degree Radian 3D lines
Distance between two points and  Line passing through two points and  A point P(x, y, z) is on the line L if and only if the direction numbers determined by P0 and P1 are proportional to those determined by P1 and P2. If the proportionality constant is t we see that the conditions are: The two points of the parametric equation of a line are: (1)
 The parametric equations of a line L through the point with direction numbers a, b and c are given by the equations: (2)
Two points formed by the equation of a line also may be written symmetrically as: Two lines with slopes of (a1, b1, c1) and (a2, b2, c2) are perpendicular
 to each other if and only if: Two lines are parallel if: Finding the distance d between 2 lines L1 and L2 that are given by the parametric equations:
 L1 L2 Step (1) Calculate the cross product of the direction numbers, the result is a vector perpendicular to both lines: Step (2) Find the norm of the vector (is a scalar value): Step (3) The unit vector in this direction is: Step (4) Find a point P on L1 where t = 0: Step (5) Find a point Q on L2 where s = 0: Step (6) Find vector PQ connecting P to Q by subtructing (Q ⎯ P):
PQ = [(x1 - x0), (y1 - y0), (z1 - z0)]
Step (7) The absolute value of the dot product of n with PQ will give the required distance d between the lines: Example 1: Find a) the parametric equations of the line passing through the points P1(3, 1, 1) and P2(3, 0, 2). b) Find a point on the line that is located at a distance of 2 units from the point (3, 1, 1). a) from equation (1) we obtain the parametric line equations: Any additional point on this line can be described by changing the value of t for example t = 2 gives the point (3, ⎯ 1, 3) which is located on the line.

b) distance from any point (x, y, z) to the point (3, 1, 1) is: Replace x, y and z by their parametric values gives: Substituting the value of t in the parametric line eqiuations yields the required point which can be located on either side of the line: Example 2: Find the equation of the line that passes through the point (1, 1, ⎯ 2) and is parallel to the line that connects the points
A(1, 2, 3) and B(2, 0, 4).
The direction numbers (values of t) of the given A B line are: The required line that passes through point (1, 1, ⎯ 2) is: Example 3: Find the distance between the lines:  Step (1) Cross product of the direction numbers is: Step (2) The norm of the vector is: Step (3) The unit vector in the line direction: Step (4) A point P on L1 where t = 0 is at: (3, ⎯ 2, 5)
Step (5) A point Q on L2 where s = 0 is at: (3, 2, ⎯ 1)
Step (6) (Q ⎯ P) = (0, 4 ⎯ 6)
Step (7) Finally the distance between the lines is: Direction angles, direction cosines and direction numbers A line has two sets of direction angles according to the pointing direction of the line If α, β, γ are the direction angles of a line then the direction cosines
 are: and: If d is the length of the line then the direction cosines values are: The direction numbers are the length of the line projected on the 3 axes x, y and z and their values are a, b and c.