Line in 3D space

Line 3D geometry - calculator

Line defined by 2 points	Point P₁(x₁,y₁,z₁)	(	,	, )
Line defined by 2 points	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 angles Direction cosines

3D lines summary Direction angles Example 1 Example 2 Example 3

3D lines summary

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 subtracting (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:

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.

3D lines - Example 1

Question:

Find (a) the parametric equations of the line passing through the points P₁(3, 1, 1) and P₂(3, 0, 2). and also find (b) a point on the line that is located at a distance of 2 units from the point (3, 1, 1).

Solution:

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 equations yields the required point which can be located on either side of the line:

3D lines - Example 2

Question:

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).

Solution:

The direction numbers (values of t) of the given A B line are:

The required line that passes through point (1, 1, ⎯ 2) is:

3D lines - Example 3

Question:

Find the distance between the lines: L1: x = 3 + 4t, y = − 2, z = 5 − 2t and the line
L2: x = 3 + s, y = 2 + 2s, z = − 1 + 7t.

Solution:

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 is:

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: