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BAAP OF ALL FORMULA LISTS
FOR IIT JEE
3D GEOMETRY
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SL# FORMULA
1
Distance Between two points
2
Dividing a Line Segment in the Ratio
,
,
.
3
Midpoint of a Line Segment
,
,
.
4
Area of a Triangle The area of a triangle with vertices
, and is given by
.
5 Volume of a Tetrahedron
The volume of a tetrahedron with vertices and is given by
Note: We choose the sign (+) or () so that to get a positive answer for volume.
d = AB = √( x 2 − x 1 )
2
2
2
λx 0 =
x 1 + λx 2
1 + λ
y 0 =
y 1 + λy 2
1 + λ
z 0 = , whereλ = , λ ≠ − 1
z 1 + λz 2
1 + λ
AC
CB
x 0 =
x 1 + x 2
2
y 0 = ( y 1 + y 2 ) / 2
z 0 = , λ = 1
z 1 + z 2
2
P 1 ( x 1 , y 1 , z 1 ), P 2 ( x 2 , y 2 , z 2 ) P 3 ( x 3 , y 3 , z 3 )
S =
y 1 z 1 1
y 2 z 2 1
y 3 z 3 1
2
z 1 x 1 1
z 2 x 2 1
z 3 x 3 1
2
x 1 y 1 1
x 2 y 2 1
x 3 y 3 1
2
1
2
P 1 ( x 1 , y 1 , z 1 ), P 2 ( x 2 , y 2 , z 2 ), P 3 ( x 3 , y 3 , z 3 ) P 4 ( x 4 , y 4 , z 4 )
V = ±
x 1 y 1 z 1 1
x 2 y 2 z 2 1
x 3 y 3 z 3 1
x 4 y 4 z 4 1
, or V = ±
x 1 − x 4 y 1 − y 4 z 1 − z 4
x 2 − x 4 y 2 − y 4 z 2 − z 4
x 3 − x 4 y 3 − y 4 z 3 − z 4
6
General Equation of a Plane
7
Normal Vector to a plane
The vector is normal to the plane
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8
Particular Cases of the Equation of a plane
If , the plane is parallel to the xaxis.
If , the plane is parallel to the yaxis.
If the plane is parallel to the zaxis.
If , the plane lies on the origin.
If , the plane is paralle to the xyplane,
If the plane is parallel to the yzplane.
If the plane is parallel to the xzplane.
9
Point Direction form
where the point lies in the plane, and the vector (A,B,C) is normal to the plane.
10
Intercept form
11
Three Point Form
12
Normal Form
where p is the perpendicular distance from the origin to the plane, and are the direction cosines of any line normal to the plane.
13
Parametric form
where are the coordinates of any unknown point on the line the
point lies in the plane, the vectors and are parallel to the plane.
14 Dihedral Angle Between Two Planes
Ax + By + Cz + D = 0
n ( A , B , C ) Ax + By + Cz + D = 0
Ax + By + Cz + D = 0
A = 0
B = 0
C = 0
D = 0
A = B = 0
B = C = 0
A = C = 0
A ( x − x 0 ) + B ( y − y 0 ) + C ( z − z 0 ) = 0 P ( x 0 , y 0 , z 0 )
x
a
y
b
z c ∣ ∣ ∣ ∣ ∣
x − x 3 y − y 3 z − z 3
x 1 − x 3 y 1 − y 3 z 1 − z 3
x 2 − x 3 y 2 − y 3 z 2 − z 3
= 0, or
x y z 1
x 1 y 1 z 1 1
x 1 y 1 z 1 1
x 2 y 2 z 2 1
x 3 y 3 z 3 1
cos α + y cos β + z cos gama − p = 0 cos α , cos β , cos γ
x = x 1 + a 1 s + a 2 t
y = y 1 + b 1 + b 2 t
z = z 1 + c 1 s + c 2 t
( x , y , z )
P ( x 1 , y 1 , z 1 ) ( a 1 , b 1 , c 1 ) ( a 2 , b 2 , c 2 )
.
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24
Parallel Lines
Two lines are parallel if.
25
Perpendicular Lines
Two lines are perpendicular if , or
26
Intersection of two lines Two Lines
and intersect at
27
Parallel Line and Plane
The straight line and the plane are
parallel iif , or
28
Perpendicular Line and Plane
The straight line and the plane are
perpendicular if or.
29
General Quadratic Equation
30
Real Ellipsoid (Case 1)
31
Imaginary Ellipsoid (Case 2)
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32 Hyperboloid of 1 Sheet (Case 3)
cos ϕ = =
s (^) 1.
s (^) 2
∣ ∣
s (^) 1 ∣∣. ∣∣
s (^) 2 ∣∣
a 1 a 2 + b 1 b 2 + c 1 c 2
√ a^21 + b^21 + c^21. √ a^22 + b^22 + c^22
s (^) 1 ∣ ∣
s (^) 2 , or = =
a 1
a 2
b 1
b 2
c 1
c 2
s (^) 1.
s (^) 2 = 0 a 1 a 2 + b 1 b 2 + c 1 c 2 = 0
x − x 1
a 1
y − y 1
b 1
z − z 1
c 1
x − x 2
a 2
y − y 2
b 2
z − z 2
c 2
x 2 − x 1 y 2 − y 1 z 2 − z 1
a 1 b 1 c 1
a 2 b 2 c 2
x − x 1
a
y − y 1
b
z − z 1
c
Ax + By + Cz + D = 0
→ n.
s = 0 Aa + Bb + Cc = 0
x − x 1
a
y − y 1
b
z − z 1
c
Ax + By + Cz + D = 0
n ∣ ∣
s = =
A
a
B
b
C
c
Ax
2
2
2
- 2 F yz + 2 Gzx + 2 Hxy + 2 P x + 2 Qy + 2 Rz + D = 0
x^2
a^2
y^2
b^2
z^2
c^2
x
2
2
y^2
b^2
z^2
c^2
33
Hyerboloid of 2 Sheets (case 4)
34
Real Quadric Cone (Case 5)
35
Imaginary Quadric Cone (case 6)
36
Elliptic Paraboloid (Case 7 )
37
Hyperbolic Paraboloid (Case 8)
38
Real Ellipstic Cylinder (case 9)
39
Imaginary Elliptic Cylinder (case 10)
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40
Hyperbolic Cylinder (Case 11)
41
Real Intersecting Planes (Case 12)
42
Imaginary Intersecting Planeks (case 13)
43 Parabolic Cylinder (Case 14)
x^2
a^2
y^2
b^2
z^2
c^2
x^2
a^2
y^2
b^2
z^2
c^2
x^2
a^2
y^2
b^2
z^2
c^2
x^2
a^2
y^2
b^2
z^2
c^2
x^2
a^2
y
2
b^2
− − z = 0
x^2
a^2
y^2
b^2
x^2
a^2
y^2
b^2
x^2
a^2
y^2
b^2
x^2
a^2
y^2
b^2
x^2
a^2
y
2
b^2
x^2
a^2
y
2
b^2
Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage (G.Tewani),
Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana, VidyaMandir
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