3D Geometry

3D-Geometry

Straight line in three dimensions

A straight line in space is uniquely determined if

(i) It passes through a given point and has a given direction, or

(ii) It passes through two given points.


Equation of a straight line through a given point A and parallel to a given vector .Cartesian Form:

Let the coordinates of A be (x₁, y₁, z₁) and the vector be . Let P(x, y, z) be an arbitrary point on L. Then and

and hence equation (1) gives

= .

Comparing the coefficients of we get

x = x₁ + b₁, y = y₁ + b₂, z = z₁ +b₃.

These are the parametric equations of line L. Eliminating the parameter, we get

... (2)

These are the Cartesian equations of the line. Equations (2) are also called the symmetrical form of the straight line, and represent a straight line through the given

point (x₁, y₁, z₁), whose direction ratios are b₁, b₂ and b₃.

For = 0, (1) gives and its Cartesian form is .... (3)

In (2) and (3), b₁, b₂, b₃ are the direction ratios of the straight line.
Equations of the straight line, through the point P(x₁, y₁, z₁), with direction cosine l, m, n are .

The general coordinates of a point on a line are given by (x₁ + lr, y₁ + mr, z₁ + nr) where r is the distance between point (x₁, y₁, z₁) and the point whose

coordinates are to be written.


Equation of a straight line through two given points

Let A (x₁, y₁, z₁) and B (x₂, y₂, z₂) be two given points and let their position vectors be respectively. That is .

Let P (x, y, z) be an arbitrary point on the line L, with position vector .

From the figure, we have .

Hence the required line passes through A and is parallel to AB i.e., and its vector equation, thus, is (r - a) = l (b - a) ....(4)

or, , .... (5)






Cartesian Form:

From (4), we have

.

Comparing the coefficients of , we obtain

(x - x₁) =(x₂ - x₁), y - y₁ = (y₂ - y₁), z - z₁ = (z₂ - z₁)

or . .... (6)

These are the Cartesian equations of the line passing through the points (x₁, y₁, z₁) and (x₂, y₂, z₂).



Collinearity of three points
Let A, B, C be three given points with position vectors respectively. The equation of the line through A and B is

.

If A, B, C are collinear, then C lies on AB

. .... (7)

This represents a relation between the position vectors of the points A, B, C, which are collinear. We note that the algebraic sum of the coefficients of is

1 - + - 1 = 0.

Let us now assume that any three vectors satisfy the relation

with k₁ + k₂ + k₃ = 0.

Dividing by k₃ ( 0), we get

with

so that

, where = -.

Hence C with position vector is collinear with A and B. We conclude that three points are collinear if there exists a linear relation between them, such that the

sum of the coefficients in it is zero.


Shortest distance between two lines

Consider two lines L₁ and L₂ in space.

then PQ = AB cos =

Note:

1. The lines L₁ and L₂ will intersect if and only if the shortest distance between them is zero i.e. PQ = 0 or .

or




2. If the lines L₁ and L₂ are respectively

and ,

the condition above for intersecting lines becomes

.


3. The lines L₁ and L₂ are skew if .and then the shortest distance between L₁ and L₂ is


.


4. If any straight line is given in general form then it can be transformed into symmetrical form and we can proceed further.