This document defines direction cosines and ratios of a line, and discusses how to find them given information about the line. It also defines planes and their equations in different forms, including the normal form using distance from origin and direction cosines of the normal vector, and the form passing through a point perpendicular to a given direction. It further discusses finding the angle between lines or between a line and plane.
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Power point presentation based on trigonometry, easy to understand, for class XI, good for learning faster and easier, also could be understood by below class XI.
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This is a slide of My talk at Kyoto Nonclassical Logic Workshop (19, November, 2015). This is based on my paper "A constructive naive set theory and infinity" which was accepted to Notre Dame Journal of Formal Logic.
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
The data is present below the pictures so as to edit it as per your needs. I wanted to use good fonts and this was the only way i could do it as the fonts would not be available on your computer.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
1. CLASS XII THREE DIMENSIONAL GEOMETRY
DIRECTION COSINES& DIRECTION RATIOS OF A LINE
The direction cosines of a line are defined as the direction cosines of any vector whose support is a given line. If
, , ,α β γ are the angles which the line l makes with the positive direction of x-axis, y-axis & z- axis
respectively,then its direction cosines are cosα , cos β , cos γ .
Or -cosα , -cos β , -cos γ .
Therefore, if l, m, n are D.C of a line,
then -l,-m,-n are also its D.C & we always have 222
nml ++ =1.
DIRECTION RATIOS OF LINE:-
Any three numbers which are proportional to the D.C of a line are called D .R of a line.
If l ,m ,n are D.C and a, b ,c are D.R of a line then a=λ l, b λm, c=λn.
TO FIND DIRCTION COSINES OF A LINE FROM ITS DIRECTION RATIO’S
Let <a, b ,c> be the D.R of a line L and <l ,m ,n>be its D.C then a=λl, b=λ m, c=λn. For some λ (≠0)
l=a/λ, m=b/λ, n=c/λ
As 222
nml ++ =1 =>
2 2 2
2 2 2
1
a b c
λ λ λ
+ + =
λ = 2 2 2
a b c± + +
2 2 2
a
l
a b c
= ±
+ +
, 2 2 2
b
m
a b c
= ±
+ +
, 2 2 2
c
n
a b c
= ±
+ +
DIRECTION RATIOS OF A LINE PASSING THROUGH TWO POINTS
The D.C. of a line joining two points P( 1 1 1, ,x y z ) &Q ( 2 2 2, ,x y z ) are
2 1 2 1 2 1
, ,
x x y y Z Z
PQ PQ PQ
− − −
< >
Where PQ= 2 2 2
2 1 2 1 2 1( ) ( ) ( )x x y y z z− + − + −
Direction Ratios of a line joining the points P( 1 1 1, ,x y z ) & Q( 2 2 2, ,x y z ) are
2 1 2 1 2 1, ,x x y y z z< − − − >
EQUATION OF A LINE IN A SPACE
EQUATION OF A LINE PASSING THROUGH A GIVEN POINT AND PARALLEL TO A GIVEN
VECTOR
Vector form: Let the line passing through the given point A with position vector a
→
and let it be parallel to vector b
→
. i.e. . AP bλ
→ →
=
BUT AP OB OA
→ → →
= −
b r aλ
→ → →
⇒ = − ⇒ r a bλ
→ → →
= + , this is vector equation of a line.
Cartesian form: Let the given point be A(( 1 1 1, ,x y z ) and
<a, b ,c> be the direction ratio & the point P( 1 1 1, ,x y z ), then 1 1 1x x y y z z
a b c
− − −
= = is symmetrical form
of line.
2. EQUATION OF A STRAIGHT LINE PASSING THROUGH TWO GIVEN POINTS
Vector form: ( )r a b aλ
→ → → →
= + −
Cartesian form:
1 1 1
2 1 2 1 2 1
x x y y z z
x x y y z z
− − −
= =
− − −
ANGLE BETWEEN TWO LINES: Let 1 2&L L be two lines passing through the origin and with
D.R. 1 1, 1,a b c & 2 2 2, ,a b c . Let P be a point on 1L & Q on 2L
Therefore the angle θ is given by
1 2 1 2 1 2
2 2 2 2 2 2
21 1 1 2 2
| |
a a b b c c
Cos
a b c a b c
θ
+ +
=
+ + + +
Vector form: Let the vectors equation of two lines be
1 1r a bλ
→ → →
= + &
2 2r a bµ
→ → →
= +
Cosθ = 1 2
1 2
.
| || |
b b
b b
→ →
→ →
Condition of perpendicularity: If the lines 1b
→
and 2b
→
are perpendicular then 1 2.b b
→ →
=0
Condition of parallelism: If the lines 1b
→
and 2b
→
are parallel then 1b
→
=λ 2b
→
Cartesian form: Let the Cartesian equation of two lines be
1 1 1
1 1 1
x x y y z z
a b c
− − −
= = &
1 1 1
2 2 2
x x y y z z
a b c
− − −
= = then
1 2 1 2 1 2
2 2 2 2 2 2
21 1 1 2 2
a a b b c c
Cos
a b c a b c
θ
+ +
=
+ + + +
Condition of perpendicularity: 90θ = g
i.e. 1 2 1 2 1 2a a bb c c+ +
Condition of parallelism: 0θ = i.e 1 1 1
2 2 2
a b c
a b c
= =
SHORTEST DISTANCE
Vector form: Let
1 1r a bλ
→ → →
= + &
2 2r a bµ
→ → →
= +
be two non interesting lines. Then the shortest distance
between the given lines is equal to
1 2 2 1
1 2
( ).( )
| |
| |
b b a a
b b
→ → → →
→ →
× −
×
Cartesian form: Let the lines be
1 1 1
1 1 1
x x y y z z
a b c
− − −
= = and
1 1 1
2 2 2
x x y y z z
a b c
− − −
= =
Shortest distance =
2 1 2 1 2 1
1 1 1
2 2 2
2 2 2
1 2 2 1 1 2 2 1 1 2 2 1( ) ( ) ( )
x x y y z z
a b c
a b c
b c b c c a c a a b a b
− − −
− + − + −
3. Note: If the lines are intersecting ⇒ lines are Coplanar
⇒ S.D = 0 ⇒ ( 1 2 2 1).(b b a a
→ → → →
× − ) = 0
or
2 1 2 1 2 1
1 1 1
2 2 2
x x y y z z
a b c
a b c
− − −
=0
SKEW LINES :
Two straight lines in space which are neither parallel nor intersecting are called Skew lines.
SHORTEST DISTANCE BETWEEN TWO PARALLEL LINES
The shortest distance between two parallel lines
1r a bλ
→ → →
= +
&
2r a bµ
→ → →
= + is given by
d = 2 1(a -a ) b
|b|
→ → →
→
×
PLANES
A Plane is a surface such that if any two distinct points are taken on it then the line containing these
points lie completely in it. i.e. every point of the line in it. Or in short A line in the space is called a plane.
NOTE: A plane is determined uniquely if any one of the following is known:
a) The normal to the plane and its distance from the origin is given.
i.e. equation of plane in normal form.
b) It passes through a point and is perpendicular to given direction
c) It passes through three non collinear points
DIFFERENT FORMS OF EQUATION OF PLANES:
EQUATION OF PLANE IN NORMAL FORM:
Let the Plane ABC be at a distance d from the origin. ON is the normal to the plane in direction n
∧
.
Equation of plane is r
→
.n
∧
=d where d= | |n
→
p
If l, m, n are the direction cosines of the normal to the plane which is at distance d from origin.
The equation of plane is lx +my +nz =d
NOTE: general form of equation of plane are r
→
. N
→
=D & Ax +By +Cz +D=0
EQUATION OF A PLANE PASSING THROUGH A GIVEN POINT & PERPENDICULAR TO A
GIVEN DIRECTION
Vector form: ( )r a
→ →
− . n
→
=0
Cartesian form: 1 1 1( ) ( ) ( ) 0A x x B y y C z z− + − + − =
4. PLANES THROUGH THE INTERSECTION OF TWO PLANES
Vector form :Let 1p and 2p be two planes with equations
1 1.r n d
→ ∧
= and 2 2.r n d
→ ∧
= . Then equation of plane passing through
the intersection of two planes is 1 2 1 2.( )r n n d dλ λ
→ → →
+ = +
Cartesian form: let 1p and 2p be two planes with equations 1 1 1 1 1 0p a x b y c z d= + + + = &
2 2 2 2 2 0p a x b y c z d= + + + = be two intersecting planes, then 1 2 0p pλ+ = represent a family of planes.
EQUATION OF PLANE PASSING THROUGH 3 NON COLLINEAR POINTS
Vector form: let a plane passing through three given opoints A,B,C with positions vectors a
→
, b
→
, c
→
. Then
equation of plane is ( )r a
→ →
− . ( ) ( ) 0b a c a
→ → → →
− × − =
Cartesian form: Let the plane pass through the points A ( 1 1 1, ,x y z ), B 2 2 2( , , )x y z ,C 3 3 3( , , )x y z .
be any point. Let P( , , )x y z be any point. Then equation of plane is
1 1 1
2 1 2 1 2 1
3 1 3 1 3 1
x x y y z z
x x y y z z
x x y y z z
− − −
− − −
− − −
=0
INTERCEPT FORM OF THE EQUATION OF PLANE:
The equation of plane in intercept form is 1
x y z
a b c
+ + =
Intersection of two planes: Let 1p and 2p be two intersecting planes with equations 1 1.r n d
→ ∧
= and
2 2.r n d
→ ∧
= .and a
→
be the position vector of any point common to them.
r a bλ
→ → →
= + where λ is real number is the vector equation of straight line.
NOTE: Whenever two planes intersect, they always intersect along a straight line.
ANGLE BETWEEN TWO PLANES: The angle between planes is defined as the angle between their
normals. If 1n
→
and 2n
→
are normals to the planes and θ be the angle between planes 1 1.r n d
→ ∧
= and
2 2.r n d
→ ∧
= . Then 1 2
1 2
.
| || |
n n
Cos
n n
θ
→ →
→ →
=
NOTE: The planes are perpendicular to each other if 1 2.n n
→ →
=0 and parallel if 1 2n n
→ →
P .
Cartesian form: Let θ be the angle between the planes 1 1 1 1 0a x b y c z d+ + + = and
2 2 2 2 0a x b y c z d+ + + = then 1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
a a bb c c
Cos
a b c a b c
θ
+ +
=
+ + + +
NOTE: Two planes are perpendicular if 0
=90θ . i.e. 1 2 1 2 1 2a a bb c c+ + =0
5. Two planes are parallel if
1 1 1
2 2 2
a b c
a b c
= =
DISTANCE OF A POINT FROM A LINE:
Vector form: The length p of the perpendicular drawn from the point p with position vector a
→
to the
plane r
→
. n
→
=d is given by p=
| . |
| |
a n d
n
→ →
→
−
NOTE: the length of perpendicular from origin to plane r
→
. n
→
=d is given by p=
| |
| |
d
n
→
Cartesian form: The length p of the perpendicular drawn from the point P( , , )x y z to the plane
Ax+By+Cz+D=0 is given by p=
1 1 1
2 2 2
Ax By Cz D
A B C
+ + +
+ +
ANGLE BETWEEN A LINE AND A PLANE:
If the equation of line is r a bλ
→ → →
= + and equation of plane is r
→
. n
→
=d .
Then the angle θ between line and normal to plane is
.
| || |
b n
Cos
b n
θ
→ →
→ →
=
So angle φ between line and plane is 90-θ .i.e. (90 )Sin Cosθ θ− =
i.e. Sinφ =
.
| || |
b n
b n
→ →
→ →
NOTE: If we have Cartesian form change it into vector form.
6. Two planes are parallel if
1 1 1
2 2 2
a b c
a b c
= =
DISTANCE OF A POINT FROM A LINE:
Vector form: The length p of the perpendicular drawn from the point p with position vector a
→
to the
plane r
→
. n
→
=d is given by p=
| . |
| |
a n d
n
→ →
→
−
NOTE: the length of perpendicular from origin to plane r
→
. n
→
=d is given by p=
| |
| |
d
n
→
Cartesian form: The length p of the perpendicular drawn from the point P( , , )x y z to the plane
Ax+By+Cz+D=0 is given by p=
1 1 1
2 2 2
Ax By Cz D
A B C
+ + +
+ +
ANGLE BETWEEN A LINE AND A PLANE:
If the equation of line is r a bλ
→ → →
= + and equation of plane is r
→
. n
→
=d .
Then the angle θ between line and normal to plane is
.
| || |
b n
Cos
b n
θ
→ →
→ →
=
So angle φ between line and plane is 90-θ .i.e. (90 )Sin Cosθ θ− =
i.e. Sinφ =
.
| || |
b n
b n
→ →
→ →
NOTE: If we have Cartesian form change it into vector form.