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.


Cartesian Form:

Equations above is 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, so and its Cartesian form is In equations above, 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
Cartesian Form:

This is 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.
, 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.
Three points with position vectors will be collinear if the vectors are parallel. We may also use the section formula. If one of the points divides the line segment, joining the remaining two, in a definite ratio, then the given points with position vectors are collinear. In the present case, let 2 =
which also satisfies 3 = .


Angle between two lines Cartesian Form:

If A(x₁, y₁, z₁), C (x₂, y₂, z₂) be the two given points and be the given vectors. The angle between the lines is given by
cos = .
If l₁, m₁, n₁; l₂, m₂, n₂ are the direction cosines of lines, then cos = l₁l₂ + m₁m₂ + n₁n₂.
The lines are perpendicular (or orthogonal) if b₁d₁+ b₂d₂+ b₃d₃= 0 or l₁l₂ + m₁m₂ + n₁n₂ = 0.

The lines are parallel if or l₁ = l₂, m₁ = m₂, n₁ = n₂. Shortest distance between two lines
Consider two lines L₁ and L₂ in space. If the two lines do not intersect and are not parallel, these lines are called skew lines.
If the lines L₁ and L₂ are respectively
and ,
the condition for intersecting lines becomes
.
The lines L₁ and L₂ are skew if . and then the shortest distance between L₁ and L₂ is
.
If any straight line is given in general form then it can be transformed into symmetrical form and we can proceed further.

PLANE
A plane is a surface such that a straight line, through any two points on it, lies wholly on it.


Equation of a plane when normal to the plane and the distance of the plane from the origin are given (normal form) Cartesian Form:

Let P (x, y, z) be any point on the plane, so that .
The equation of the plane will be
lx + my + nz = p
or Ax + By + Cz =
which is a linear equation in x, y, z. The coefficients A, B, C are the direction ratios of the normal to the plane, whereas l, m, n are the actual direction cosines of the normal.
A plane parallel to the plane Ax + By + Cz = D has the same normal. Hence the direction cosines of the normal to the two planes are same. Hence the equation of a plane parallel to Ax + By + Cz = D is Ax + By + Cz = D₁. Here is the perpendicular distance of this plane from the origin. If D and D₁ are of opposite signs, then the two planes lie on the opposite sides with respect to the origin i.e. the origin lies between the two planes. If the plane Ax + By + Cz = D intersects the coordinates axes at the points (a, 0, 0), (0, b, 0), (0, 0, c), then and the equation of the plane simplifies to , which is called the intercept form of the equation of the plane.


Equation of a plane passing through a given point and perpendicular to a given direction Cartesian Form:

With , we get the Cartesian form as
l (x - x₁) + m (y - y₁) + n (z - z₁) = 0.
Here l, m, n are the direction cosines. In terms of the direction ratios a, b, c of the normal, this equation becomes
a (x - x₁) + b (y - y₁) + c (z - z₁) = 0
Let A (x₁, y₁, z₁), B (x₂, y₂, z₂), C (x₃, y₃, z₃) be three points on a plane. Equation of the plane through A (x₁, y₁, z₁) is (x - x₁) + b (y - y₁) + c (z - z₁) = 0 This passes through the points B and C so that
a (x₂ - x₁) + b (y₂ - y₁) + c (z₂ - z₁) = 0,
a (x₃ - x₁) + b (y₃ - y₁) + c (z₃ - z₁) = 0.
Eliminating a, b, c from these equations, we obtain the equation of the plane containing A, B, C as .


Equation of a plane passing through a point and parallel to two given lines Cartesian form:

Let the point A be (x₁, y₁, z₁) and let the direction ratios of the given lines be (b₁, b₂, b₃) and(c₁, c₂, c₃). Hence the equation of plane passing through a point and parallel to two given lines is
.

If a plane contains the line and is parallel to the line then all the points of the first line will be lying in this plane. Moreover, the plane will be parallel to both the lines. Hence the equation of the plane is
Two lines are coplanar if a plane can be drawn to contain both the lines.


Plane passing through the intersection of two given planes
In Cartesian system, the equation of a plane passing through the intersection of the planes
a₁x + b₁y + c₁z = d_1­ and a₂x + b₂y + c₂z = d₂
is a₁x + b₁y + c₁z - d₁ + l (a₂x + b₂y + c₂z - d₂) = 0.
Two planes intersect in a line. Hence the equations a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂ together represent a line.



Equation of the plane through a given line:
(i) If the equations of a line are given in general form as a₁x+b₁y+c₁z+d₁=0=a₂x+b₂y+c₂z+d₂, the equation of plane passing through this line is (a₁x + b₁y + c₁z + d₁) + l(a₂x + b₂y + c₂z + d₂) = 0.
(ii) If equations of the line are given in symmetrical form as , then equation of the plane is a(x - x₁) + b(y - y₁) + c(z - z₁) = 0, where a, b, c are given by al + bm + cn = 0.


Angle between two planes

Let and be two planes. The angle between the planes is same as the angle between their normals. Hence angle between the planes is given by
cos = .
In case, vectors along the normals (or direction ratios of the normal) are given, then
cos = .
The planes are perpendicular if = . The planes are parallel if Angle between the planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is . If a₁a₂ +b₁b₂ +c₁c₂ = 0, then the planes are perpendicular to each other.
If then the planes are parallel to each other.


Angle between a line and a plane
Let the equation of the line be and that of the plane , is a unit vector along the normal to the plane. Let us draw a plane through the given line which is perpendicular to the given plane. Let L be the line of intersection of this plane with the given plane. The angle between the given line and L defines the angle between the given line and the given plane.
Let f be the angle between the given line and the normal to the plane. Then cos = .
If is the angle between the given line and the given plane. then = sin = cos = .
If is a vector along the normal to the plane, then sin = .

If the equation of a plane is ax + by + cz + d = 0, then direction ratios of the normal to this plane are a, b, c. If the equations of the straight line are , then angle q between the plane and the straight line is given by sin q = .
Plane and straight line will be parallel if al + bm + cn = 0 Plane and straight line will be perpendicular if . The line will lie in the plane if al + bm + cn =0 and ax₁ + by­₁ + cz₁ + d = 0.


Distance of a point from a plane

Consider a point A with position vector and a plane with equation . Draw plane p through A and parallel to the given plane.

Equation of the plane p is or . From figure,
AM = NL
= ON - OL
=
Hence distance of the point A from the given plane
= |AM| =.


In case the equation of the plane is Ax + By + Cz = D or ,
the distance of from the plane is
, where .


Position of a Point w.r.t. a Plane:
Two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) lie on the same or on the opposite sides of a plane ax + by + cz + d = 0 according as ax₁ + by₁ + cz₁ + d and ax₂ + by₂ + cz₂ + d are of the same or of the opposite signs. The plane divides the line joining the points P and Q externally or internally according as P and Q are lying on the same or the opposite sides of the plane.
The distance between two parallel planes is the algebraic difference of perpendicular distances of the planes from the origin.

Bisector Planes of Angle between two Planes:
The equation of the planes bisecting the angles between two given planes a₁x +b₁y +c₁z +d₁ = 0 and a₂x + b₂y + c₂z +d₂ = 0 is
.
If angle between the bisector plane and one of the planes is less than 45° then it is acute angle bisector otherwise it is obtuse angle bisector. If a₁a₂ + b₁b₂ + c₁c₂ is negative, then the origin lies in the acute angle between the given planes provided d₁ and d₂ are of the same sign and if a₁a₂ + b₁b₂ + c₁c₂ is positive, then the origin lies in the obtuse angle between the given planes.