Three Dimensional Geometry

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1. IIT-JEE / ENGINEERING

MATHS

FORMULA

Three Dimensional Geometry

By

Assistant Professor (Computer Science)
Director, BST, Kokar
&
Assistant Professor (Computer Science)
“Asst, Director, BSTI, Kokar

Buddha S

2. Three dimensional Geometry

IMPORTANT DEFINATIONS, FORMULAE AND METHODS
1. Direction cosines of a line

(6) Direction cosines of a line ate the cosines of the angles made by the line X-

El
me ‘Y-axis and Z-axis,
El (6) Direction cosines of X-anis, Y-anis and Z-axis are respectively 1,0,0:0, 1,0

and 0,0, 1

D Fm nar te dei SoS TFA Me Y
2. Direction ratio ofaline > — 5

(6) Tnesofäre ihe-nümbers say a, He which are propos

cotée) 7

món ig tem
Sap

Let AB be a lind join AC 3,5 Jan, BC, Mas): Then the direction

ratios of ie Ting AB-are dh) 24, and its direction cosines are

[ay al Pad here JAM lO AER |
Gv) Thé direcion ratos of yetor P = af + sch/are abc.
3. Angle between two lines

ERING At Kokar, Ranchi - 834001,

GE Ayman and mi, ae the direction cosines of two lines and 8 is the

Uma ne
and @h.c, are the direction ratios of v6 lines and is the angle

bras

between them, hen C0 ==
ne

(Gi) For perpendicular Ines, LL, + mm, +n, =0

hmm

(Gv) For parallel ines,

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3. al Gi

4. Equation of a line passing through a given po
vector
(6) The vector form of a line passing through a point A(á) and paralelo given

+ AB where À ix sear

vector is given by

(6) Canesian equation of

and having direcion

ine passing through ACs, 3

ratios ab cis LE 2

5. Equation of line passing through two given points
di The vector fm oi asin ro tw pois AG) and AG) is even

by =

- 834001, Jharkhand

ara

gio Carésian edition of a ine passing. thro pain ACH

Ds nz

6. Angle between two lines in vector form)
1 O isan ange berween F<, ah, aad 7 = d+ a then 4

7: Skewlines |.

7) Tio lines in a space Which are either parallel nor intersecting are called skew
fines. 5 =

‘Angle between skew lines is the angle between two intersecting lines drawn

ny point (preferably through the origin) paralelo €ach ofthe skew lines.

8. Shortest distance between two skew lines À

given by,

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ometry

4. Jharkhand

ERING At Kokar, Ranchi - 834001,

he,

lex @-al G ha
ing A.
2. Const Seto genes to ee

Condition for wo tines EU PEN and

y a a

EP

(6) Inerscctis

(donor infenestis || a
i a

10, Equation o plane nthe normal form
Inthe veer form, equation of plane whichis ata distance“ from he origin
ani a uni yecor normal tothe given plan, direc rom the origin to the

planes
Fñ=d
(6) Equation oFa plane hich is at distance: from the origin and 1,m mate
the direction cosines oP Uh normal othe panes
Icempene=d

(6) The equation of a plane passing through a point A(@) and perpendicular o the

given vector iis given by AAA À

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5. Gv) The equation of a plane passing through a point Al;

perpendicular tothe given line with direction ratios a,b, s given by

axa) +H Ye
11, General equation of a plane
General equation ofa plane having a,b. e as direction ratios of the normal to the

a

Plane is ax-+by

Jharkhand

12. Method to reduce general form of equation to the normal form
Let the general equ

ofthe plane sex, + by 4 67 + d 20

> -ax-by send

Diving ASF (TEE

Loa E 15 È u
21 many,
TEE pipa TTL ON
À mana à ye pul fg plane RS D ges ATE.
‘where the constant K is determined by-a given eondition. a

i) The equation of pláne paralelo te plane ax bye }d = 08 given by

ERING At Kokar, Ranchi - 834001,

aby +ez+ K =O) where Kis determined by ven condition

14, Equation of plane through the interseetion of to planes
() Veclor equation of «plane ha passes rg the intersection of to plans

lana FR, de given bY AO FA) de Ad. where jis) any

parameter ae
(6) Canesan equation of a plane that passes through the Intersection oF two planes
axthy razed, =Oand — acrbyrectd,=0 is given by

Cathy + dz rd Aaa et #4

15. Equation of plane passing through three given points
@ 18 through three non collinear points

here A is any parameter,

The vector equation of a plane pas

Aa. 26) a8 Ci ghen by @-H[ 6-2) XG-2)]=0

(6) Cantesian equation of a plane passing through three non collinear points

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6. - 834001, Jharkhand

al Geometry

A Bas is gen by

16. Equation of a plane passing through a given point and parallel to two given
lines
‘through a given point AG) and parallel

(6) The vector equation ofa plane pass

tothe given veces had E O)

(69 Cani dquaton Fa plane passing trough ghE BRIN AG
Bald aden is sven by

paralelo 16 given lines having direction

= fs ve
LL SrA?

AT. Angle between (wo planes:
‚The angle between two given pines isthe angle between their normal.

iy the Vector form, ¡f-0-¡S he angle Ben the two planes A5 =a, an

Faden one

ln, the Cartesian om, if @-is the siglo between the two) planes

cos.

18. Angle between a line and a plane
(6) The angle between a line ada plane the complement of the angle between
{he line and the normal To ie plane. Z

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7. Jharkhand

ERING At Kokar, Ranchi - 834001,

Gi) In Cartesian form, if 8 is the angle between the line SL o ZN EL
ah

sind

19. Coplanarity of two lines

o s+ 7b, are coplanar if

&

(iy M Can fm two — panes M hy reed =0 and
re en no

a (BS rd, -Oireeoplnarit a, dy

N ap) by

2

‘planarity of two lines
Gy In vector form, two lines 7 =d, + dh and red, + fe coplanar if
0

(a, Ah, X 5,

In Cartesian ¿form to Ces

a ar copiar ia,

21, Length of perpendicular from a point to a plane.
No gen eng of paper rn pole 40 hei

is giventy ©

y

(6) In Caresian form, te length of the perpendicular dei from A(x,

Foy tee tal

apres

nby

the plane ar by eo

22. Equation ofa plane in he intercept form
Equation of a plane that makes intercepts of lengths a,b, € with the X-axis, Y-axis

and Z-axis respectively i given by À

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