Neet class 11 12 basic mathematics notes

ChhaviSamriya 181 views 86 slides Nov 09, 2024
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About This Presentation

Akash notes


Slide Content

Welcome to
Orientation Sessions
NOTES

Functions
•Input variable is independentand output variable is
dependenton it.
•A function is a mathematical relationship between
independentand dependent variables
Algebraic Trigonometric Exponential Logarithmic
Functions

Polynomials
�=��=�
0+�
1�
1
+�
2�
2
+�
3�
3
+…….�
��
�
Constants
Variables
= =
Degreeofa
polynomial
Highestpower
Maximum
number
ofsolutions
Algebraic Function

The graphical representation of a linear function is a straight line
1
Linear Function
�=��=�
0+�
1�
1
Constants
Variables
= = =
Degreeofa
polynomial
Highestpower
Maximum
number
ofsolutions
Equation of line
�=��+�
�→�-intercept of line
=tan??????=
�2−�1
�2−�1
�→Slope of line

Rahul is standing 4�East and 5�North from Ashok.
Taking Ashok’s position as origin, represent Rahul’s
position on the coordinate plane given below.
�(�)
�(�)
Coordinates
of Rahul’s
position

Ashok has now moved 1�East and 1�North while
Rahul moved 1�South. Calculate shortest distance
between Rahul and Ashok.
Shortest distance =�
2−�
1
2
+�
2−�
1
2
Shortest distance =4−1
2
+4−1
2
Shortest distance =3
2
+3
2
Shortest distance =32�
�
�
1
1
Rahul→(4,4)
Ashok→(1,1)
1

Determine direction of Rahul’s position (4,4)with
respect to Ashok (1,1).
Slope/Direction=tan??????=
�
2−�
1
�
2−�
1
tan??????=
4−1
4−1
=1
tan??????=tan45°
??????=45°from +���−axis
�
�
Rahul→(4,4)
Ashok→(1,1)
??????

Case ii: c=0
Passing through origin
�
�=��=−�
�
Case i: �=0
Parallel to x-axis
�=+�
(0,�)
�
�
�
Special cases

�=��+�
Straight line equation
Slope -intercept
�
�
??????
�
�−�
1=�(�−�
1)
Point -slope
�
�
??????
(�
1,�
1)
�−�
1=
(�
2−�
1)
(�
2−�
1)
(�−�
1)
(�
2,�
2)
(�
1,�
1)
Two-Point

�=��
�
+��+�
(�≠0)
•Graph of �=��
2
+��+�is
always parabolic.
•If �>0, then parabola opens
upward.
•If a<0, then parabola opens
downward.
Quadratic function:
Root 1 :
Root 2 :
&#3627409148;=
−&#3627408463;−&#3627408463;
2
−4&#3627408462;&#3627408464;
2&#3627408462;
&#3627409149;=
−&#3627408463;+&#3627408463;
2
−4&#3627408462;&#3627408464;
2&#3627408462;
&#3627408462;&#3627408485;
2
+&#3627408463;&#3627408485;+&#3627408464;=0
Sum of roots
Difference of roots
Product of roots
&#3627409148;+&#3627409149;=−
&#3627408463;
&#3627408462;
&#3627409149;−&#3627409148;=
&#3627408463;
2
−4&#3627408462;&#3627408464;
&#3627408462;
&#3627409148;&#3627409149;=
&#3627408464;
&#3627408462;
Equation in terms of roots
&#3627408485;
2
−&#3627409148;+&#3627409149;&#3627408485;+&#3627409148;&#3627409149;=0
Quadratic Function

&#3627408486;=&#3627408462;&#3627408485;
2
+&#3627408464;
Parabola(Quadratic Curve)
&#3627408486;=&#3627408462;&#3627408485;
2
&#3627408486;=&#3627408462;&#3627408485;
2
&#3627408462;>0

Plane Angle in Radian
??????=
&#3627408514;&#3627408531;&#3627408516;??????&#3627408518;&#3627408527;&#3627408520;&#3627408533;&#3627408521;
&#3627408531;&#3627408514;&#3627408517;&#3627408522;&#3627408534;&#3627408532;&#3627408528;&#3627408519;&#3627408516;&#3627408522;&#3627408531;&#3627408516;??????&#3627408518;
=
??????
&#3627408531;
(&#3627408531;&#3627408514;&#3627408517;)
System -I
Degree (°)
Minute (′)
Second (′′)
System -II
Radian (&#3627408479;&#3627408462;&#3627408465;)
1°=60′
1′=60′′
1°=
2??????
360
&#3627408479;&#3627408462;&#3627408465;
&#3627408522;)540°=
2??????×540
360
=3??????&#3627408479;&#3627408462;&#3627408465;
&#3627408465;&#3627408466;&#3627408468;&#3627408479;&#3627408466;&#3627408466;&#3627408481;&#3627408476;&#3627408479;&#3627408462;&#3627408465;&#3627408470;&#3627408462;&#3627408475;
×
2??????
360
&#3627408479;&#3627408462;&#3627408465;&#3627408470;&#3627408462;&#3627408475;&#3627408481;&#3627408476;&#3627408465;&#3627408466;&#3627408468;&#3627408479;&#3627408466;&#3627408466;
×
360
2??????
&#3627408470;&#3627408470;)210°=
2??????×210
360
=
7
6
??????&#3627408479;&#3627408462;&#3627408465;
&#3627408470;&#3627408470;&#3627408470;)270°=
2??????×270
360
=
3
2
??????&#3627408479;&#3627408462;&#3627408465;

Trigonometric Ratios
csc??????=
1
sin??????
sec??????=
1
cos??????
cot??????=
1
tan??????
sin??????=
&#3627408476;&#3627408477;&#3627408477;
ℎ&#3627408486;&#3627408477;
cos??????=
&#3627408462;&#3627408465;&#3627408471;
ℎ&#3627408486;&#3627408477;
tan??????=
&#3627408476;&#3627408477;&#3627408477;
&#3627408462;&#3627408465;&#3627408471;
IfA+B=90°,then
&#3627409359;.sin&#3627408488;=cos&#3627408489;&#3627409360;.tan&#3627408488;=cot&#3627408489;&#3627409361;.sec&#3627408488;=csc&#3627408489;

Sine, Cosine and Tangent Curve
Graph of &#3627408486;=sin&#3627408485;
The period of sin&#3627408485;is 2??????.
−1<sin&#3627408485;<+1
Graph of &#3627408486;=cos&#3627408485;
The period of cos&#3627408485;is 2??????.
−1<cos&#3627408485;<+1
Graph of &#3627408486;=tan&#3627408485;
The period of tan&#3627408485;is ??????.
tan&#3627408485;??????(−∞,∞)

sin(&#3627408436;±&#3627408437;)=sin&#3627408436;cos&#3627408437;±cos&#3627408436;sin&#3627408437;
cos(&#3627408436;±&#3627408437;)=cos&#3627408436;cos&#3627408437;∓sin&#3627408436;sin&#3627408437;
tan(&#3627408436;±&#3627408437;)=
tan&#3627408436;±tan&#3627408437;
1∓tan&#3627408436;tan&#3627408437;
Additive Multiple angle
sin2&#3627408436;=2sin&#3627408436;cos&#3627408436;
cos2&#3627408436;=cos
2
&#3627408436;−sin
2
&#3627408436;=1−2sin
2
&#3627408436;=2cos
2
&#3627408436;−1
tan2&#3627408436;=
2tan&#3627408436;
1–tan
2
&#3627408436;

sin90°−??????=+cos??????
cos90°−??????=+sin??????
tan90°−??????=+cot??????
csc90°−??????=+sec??????
sec90°−??????=+csc??????
cot90°−??????=+tan??????
sin180°−??????=+sin??????
cos180°−??????=−cos??????
tan180°−??????=−tan??????
csc180°−??????=+csc??????
sec180°−??????=−sec??????
cot180°−??????=−cot??????
Summary

&#3627408467;(&#3627408485;)=&#3627408462;
&#3627408485;
Where,
•&#3627408462;→base & &#3627408462;>0&&#3627408462;≠1
•&#3627408485;→any real number &#3627408486;=&#3627408466;
&#3627408485;
Euler’s number &#3627408466;≈2.718
Special case:
Natural Exponential Function
Exponential Function

•y=log
&#3627408462;&#3627408485;is logarithmic function where &#3627408462;>0and &#3627408462;≠1
•Product Rule : log
&#3627408462;&#3627408474;&#3627408475;=log
&#3627408462;&#3627408474;+log
&#3627408462;&#3627408475;
•Quotient Rule : log
&#3627408462;
&#3627408474;
&#3627408475;
=log
&#3627408462;&#3627408474;−log
&#3627408462;&#3627408475;
•Power Rule : log
&#3627408462;&#3627408474;
&#3627408475;
=&#3627408475;log
&#3627408462;&#3627408474;
•log
&#3627408462;1=0
•log
&#3627408462;0=&#3627408482;&#3627408475;&#3627408465;&#3627408466;&#3627408467;&#3627408470;&#3627408475;&#3627408466;&#3627408465;
•log
&#3627408462;&#3627408462;=1
Logarithmic Function

Logarithmic Function
base =e,
&#3627408467;&#3627408485;=&#3627408473;&#3627408476;&#3627408468;
&#3627408466;&#3627408485;=&#3627408473;&#3627408475;&#3627408485;
base =10,
&#3627408467;&#3627408485;=&#3627408473;&#3627408476;&#3627408468;
10&#3627408485;
&#3627408462;>0
&#3627408462;≠1
&#3627408485;>0
&#3627408486;=&#3627408473;&#3627408476;&#3627408468;
&#3627408462;&#3627408485;
&#3627408485;=&#3627408462;
&#3627408486;
Common logarithmNatural Logarithm
Graph of Logarithmic Function &#3627408486;=&#3627408473;&#3627408476;&#3627408468;
&#3627408463;&#3627408485;
b > 1
0 < b <1

If log
102=0.3010and log
103=0.4771then find the
approximate value of log
1036.
log
1036=log
10(9×4)
=log
109+log
104
=log
103
2
+log
102
2
=2log
103+2log
102
=2×0.4771+2×0.3010
=1.5562
log
1036=1.5562

Summary
&#3627408474;=tan??????
Tangen
t
Secant
•Theslope of a linegives the measure
of its steepnessand direction.
•Slope is generally denoted by m.
•In the case of a secant, the line
intersects the curve at two points.
•A tangentis defined asa line
touching the circle at only one
point..
&#3627408485;
&#3627408486;
−&#3627408485;
−&#3627408486;
θ
L

Summary
&#3627408486;
&#3627408485;
&#3627408436;
Δ&#3627408485;
Δ&#3627408486;
&#3627408437;
??????
&#3627408486;
&#3627408485;
Tangent
??????
tan??????=
&#3627408486;
2−&#3627408486;
1
&#3627408485;
2−&#3627408485;
1
=
Δ&#3627408486;
Δ&#3627408485;
Change in yw.r.t change in x,
&#3627408486;
2−&#3627408486;
1→Difference in &#3627408486;
&#3627408485;
2−&#3627408485;
1→Difference in &#3627408485;
&#3627408485;=independent variable
&#3627408486;=dependent variable
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=lim
∆&#3627408485;→0
Δ&#3627408486;
Δ&#3627408485;
, Where ∆&#3627408485;→0
Slope=tan??????=
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=Differentiation of &#3627408538;w.r.t &#3627408537;

Differentiation
•Power rule for differentiation
&#3627408465;
&#3627408465;&#3627408485;
(&#3627408485;
&#3627408475;
)= &#3627408475;&#3627408485;
&#3627408475;−1
•Differentiation of any
constant is zero.
&#3627408485;=independent variable
&#3627408486;=dependent variable
&#3627408486;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
&#3627408485;
1
1
&#3627408485;
1
2&#3627408485;
4
&#3627408485;
1
4
&#3627408485;

3
4
1
&#3627408485;

1
&#3627408485;
2
&#3627408486;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
1
&#3627408485;
3

3
&#3627408485;
4
1
&#3627408485;
7

7
&#3627408485;
8
1
&#3627408485;
16

16
&#3627408485;
17
5
&#3627408485;

5
&#3627408485;
2

Trigonometric, Exponential and
Logarithmic Functions
(sin&#3627408485;)=cos&#3627408485;;
&#3627408465;
&#3627408465;&#3627408485;
(1)
(tan&#3627408485;)=sec
2
&#3627408485;;
&#3627408465;
&#3627408465;&#3627408485;
(3)
(sec&#3627408485;)=sec&#3627408485;tan&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(5)
(&#3627408464;&#3627408476;&#3627408480;&#3627408466;&#3627408464;&#3627408485;)=−cosec&#3627408485;cot&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(6)
(cot&#3627408485;)=−cosec
2
&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(4)
(cos&#3627408485;)=−sin&#3627408485;;
&#3627408465;
&#3627408465;&#3627408485;
(2)
Exponential Functions
Trigonometric
&#3627408465;
&#3627408465;&#3627408485;
&#3627408466;
&#3627408485;
=&#3627408466;
&#3627408485;
Logarithmic
&#3627408465;
&#3627408465;&#3627408485;
ln&#3627408485;=
1
&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
&#3627408464;=0
&#3627408465;
&#3627408465;&#3627408485;
&#3627408472;&#3627408485;=&#3627408472;
Constant

Summary
•Derivative of constant times a function
&#3627408465;
&#3627408465;&#3627408485;
&#3627408462;&#3627408467;&#3627408485;=&#3627408462;
&#3627408465;&#3627408467;&#3627408485;
&#3627408465;&#3627408485;
•Derivative of a sum or a difference of two functions
•Derivative of a constant term
&#3627408517;
&#3627408517;&#3627408537;
&#3627408514;=&#3627409358;
&#3627408517;
&#3627408517;&#3627408537;
&#3627408519;&#3627408537;±&#3627408520;(&#3627408537;)=
&#3627408517;&#3627408519;&#3627408537;
&#3627408517;&#3627408537;
±
&#3627408517;&#3627408520;&#3627408537;
&#3627408517;&#3627408537;

360°270°180°90°-90°-180°-270°-360°
-2π 2π3π/2ππ/2
0
-π/2-π-3π/2
0
1
0.
5
-0.5
-1
&#3627408486;=sin&#3627408485;
x
&#3627408436;
&#3627408437;
360°270°180°90°-90°-180°-270°-360°
&#3627408486;=cos&#3627408485;
x
-2π 2π3π/2ππ/20-π/2-π-3π/2
1
0.
5
-0.5
-1
0
&#3627408436;
&#3627408437;
&#3627408486;=sin&#3627408485;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=cos&#3627408485;

A
C D
2&#3627408464;&#3627408476;&#3627408480;&#3627408485;−6sectan&#3627408485;
2sin&#3627408485;−6sec&#3627408485;tan&#3627408485;2sin&#3627408485;−6sec&#3627408485;tan&#3627408485;
Differentiate f (&#3627408485;) = 2cos (&#3627408485;)−6 sec(&#3627408485;)+3
&#3627408465;
&#3627408465;&#3627408485;
cot&#3627408485;=−csc
2
&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(sec&#3627408485;)=sec&#3627408485;tan&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
csc&#3627408485;=−csc&#3627408485;cot&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(sin&#3627408485;)=cos&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(tan&#3627408485;)=sec
2
&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
cos&#3627408485;=−sin&#3627408485;
B−2sin&#3627408485;−6sec&#3627408485;tan&#3627408485;

&#3627408465;&#3627408467;&#3627408485;.&#3627408468;(&#3627408485;)
&#3627408465;&#3627408485;
=
&#3627408465;&#3627408467;&#3627408485;
&#3627408465;&#3627408485;
&#3627408468;&#3627408485;+
&#3627408465;&#3627408468;&#3627408485;
&#3627408465;&#3627408485;
&#3627408467;&#3627408485;=&#3627408467;

&#3627408485;&#3627408468;&#3627408485;+&#3627408468;

&#3627408485;&#3627408467;(&#3627408485;)
Product Rule
Example: &#3627408452;1.&#3627408486;=&#3627408485;sin&#3627408485;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=&#3627408485;sin&#3627408485;

+sin&#3627408485;(&#3627408485;)′
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=&#3627408485;cos&#3627408485;+sin&#3627408485;
&#3627408452;2.&#3627408486;=&#3627408485;
3
sin&#3627408485;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=&#3627408485;
3
sin&#3627408485;

+sin&#3627408485;(&#3627408485;
3
)′
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=&#3627408485;
3
cos&#3627408485;+3&#3627408485;
2
sin&#3627408485;
&#3627408452;3.&#3627408486;=&#3627408485;
2
&#3627408466;
&#3627408485;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=&#3627408485;
2
&#3627408466;
&#3627408485;′
+&#3627408466;
&#3627408485;
(&#3627408485;
2
)′
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=&#3627408485;
2
&#3627408466;
&#3627408485;
+2&#3627408485;&#3627408466;
&#3627408485;
&#3627408452;4.&#3627408486;=cos&#3627408485;.log
&#3627408466;&#3627408485;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=cos&#3627408485;log
&#3627408466;&#3627408485;

+log
&#3627408466;&#3627408485;(cos&#3627408485;)′
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=
cos&#3627408485;
&#3627408485;
−sin&#3627408485;.log
&#3627408466;&#3627408485;

Differentiate&#3627408467;&#3627408485;=tan&#3627408485;sec&#3627408485;w.r.t &#3627408485;.
&#3627408465;
&#3627408465;&#3627408485;
cot&#3627408485;=−csc
2
&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(sec&#3627408485;)=sec&#3627408485;tan&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
csc&#3627408485;=−csc&#3627408485;cot&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(sin&#3627408485;)=cos&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
(tan&#3627408485;)=sec
2
&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
cos&#3627408485;=−sin&#3627408485;
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=&#3627408481;&#3627408462;&#3627408475;&#3627408485;s&#3627408466;&#3627408464;&#3627408485;

+s&#3627408466;&#3627408464;&#3627408485;(&#3627408481;&#3627408462;&#3627408475;&#3627408485;)′
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=(&#3627408480;&#3627408466;&#3627408464;&#3627408485;&#3627408481;&#3627408462;&#3627408475;&#3627408485;)tan&#3627408485;+sec&#3627408485;(&#3627408480;&#3627408466;&#3627408464;&#3627408485;
2
)
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=(&#3627408480;&#3627408466;&#3627408464;&#3627408485;&#3627408481;&#3627408462;&#3627408475;&#3627408485;
2
+&#3627408480;&#3627408466;&#3627408464;&#3627408485;
3
)
A sec&#3627408485;tan
2
&#3627408485;B &#3627408485;tan
2
&#3627408485;+sec
3
&#3627408485;
C sec&#3627408485;tan
2
&#3627408485;+sec
3
&#3627408485;D sec&#3627408485;+sec
3
&#3627408485;

&#3627408465;
&#3627408465;&#3627408485;
&#3627408467;&#3627408485;
&#3627408468;&#3627408485;
=
&#3627408465;&#3627408467;&#3627408485;
&#3627408465;&#3627408485;
&#3627408468;&#3627408485;−
&#3627408465;&#3627408468;&#3627408485;
&#3627408465;&#3627408485;
&#3627408467;&#3627408485;
(&#3627408468;&#3627408485;)
2
Division Rule
Derivative of tan&#3627408485;w.r.t &#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
tan&#3627408485;=
&#3627408465;
&#3627408465;&#3627408485;
sin&#3627408485;
cos&#3627408485;
=
cos&#3627408485;.
&#3627408465;
&#3627408465;&#3627408485;
sin&#3627408485;−sin&#3627408485;.
&#3627408465;
&#3627408465;&#3627408485;
cos&#3627408485;
cos
2
&#3627408485;
=
cos
2
&#3627408485;+sin
2
&#3627408485;
cos
2
&#3627408485;
=1+tan
2
&#3627408485;=sec
2
&#3627408485;
Example:
Differentiate: &#3627408467;&#3627408485;=
1
1−&#3627408485;
2
w.r.t
to&#3627408485;using division rule.
&#3627408467;&#3627408485;=
1
1−&#3627408485;
2
&#3627408467;

&#3627408485;=
1−&#3627408485;
2
0−1−2&#3627408485;
1−&#3627408485;
22
=
2&#3627408485;
1−&#3627408485;
22


&#3627408465;&#3627408467;
&#3627408465;&#3627408485;
=
&#3627408465;&#3627408467;
&#3627408465;&#3627408468;
.
&#3627408465;&#3627408468;
&#3627408465;&#3627408485;
⇒&#3627408467;′(&#3627408485;)=&#3627408467;′(&#3627408468;)&#3627408468;′(&#3627408485;)
Example of a
Composite function
•&#3627408467;&#3627408468;=sin&#3627408468;
•&#3627408468;&#3627408485;=&#3627408485;
2
•&#3627408467;&#3627408468;&#3627408485;=sin(&#3627408485;
2
)
If &#3627408467;=&#3627408467;(&#3627408468;);&#3627408468;=&#3627408468;(&#3627408485;), then it can be
written as &#3627408467;(&#3627408468;(&#3627408485;))
Chain Rule
Differentiate: &#3627408467;&#3627408485;=&#3627408466;
sinln&#3627408485;
w.r.t to&#3627408485;using
to chain rule.
&#3627408467;&#3627408485;=&#3627408466;
sinln&#3627408485;
&#3627408467;

(&#3627408485;)=
&#3627408465;
&#3627408465;&#3627408485;
&#3627408466;
sinln&#3627408485;
=&#3627408466;
sinln&#3627408485;
.cosln&#3627408485;.
1
&#3627408485;
Differentiate: &#3627408468;&#3627408481;=
1
4&#3627408481;
2
−3&#3627408481;+2
2
w.r.t to&#3627408485;
using chain rule.
&#3627408468;&#3627408481;=
1
4&#3627408481;
2
−3&#3627408481;+2
2
&#3627408468;

(&#3627408481;)=
&#3627408465;
&#3627408465;&#3627408481;
4&#3627408481;
2
−3&#3627408481;+2
−2
=−
(16&#3627408481;−6)
4&#3627408481;
2
−3&#3627408481;+2
3

&#3627408486;=cos&#3627408485;
2
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=
&#3627408465;
&#3627408465;&#3627408485;
cos&#3627408485;
2
=−sin&#3627408485;
2
∙2&#3627408485;

&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=−2sin&#3627408485;
2

&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;+&#3627408485;.
&#3627408465;
&#3627408465;&#3627408485;
sin&#3627408485;
2

&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=−2sin&#3627408485;
2
∙1+cos&#3627408485;
2
∙2&#3627408485;∙&#3627408485;

&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=−2&#3627408485;sin&#3627408485;
2

&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=−4&#3627408485;
2
cos&#3627408485;
2
−2sin&#3627408485;
2
Successive or Double Differentiation
&#3627408486;=cos&#3627408485;
2
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=
&#3627408465;
&#3627408465;&#3627408485;
cos&#3627408485;
2
=−2&#3627408485;sin&#3627408485;
2
&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=−2
&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;sin&#3627408485;
2
=−2[sin&#3627408485;
2
+2&#3627408485;
2
cos&#3627408485;
2
]

&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=−2[sin&#3627408485;
2
+2&#3627408485;
2
cos&#3627408485;
2
]

Applications of Differentiation
Variation of one quantity
with respect to another
1
Slope of curve2
Maxima and minima
of any function
3

If the motion of a particle is represented by,
&#3627408480;=&#3627408481;
3
+&#3627408481;
2
−&#3627408481;+2&#3627408474;. Find the magnitude of
position, velocity & acceleration of the particle
at &#3627408481;=2&#3627408480;.
At &#3627408481;=2&#3627408480;
Position =12&#3627408474;Position&#3627408480;=&#3627408481;
3
+&#3627408481;
2
−&#3627408481;+2
Velocity =
&#3627408465;&#3627408480;
&#3627408465;&#3627408481;
=3&#3627408481;
2
+2&#3627408481;−1
Acceleration =
&#3627408465;
2
&#3627408480;
&#3627408465;&#3627408481;
2
=6&#3627408481;+2
Velocity =15&#3627408474;/&#3627408480;
Acceleration =14&#3627408474;/&#3627408480;
2

•for &#3627408485;
1<&#3627408485;
2, We observe &#3627408467;&#3627408485;1<&#3627408467;(&#3627408485;
2)
??????(&#3627408485;1)
??????(&#3627408485;2)
&#3627408485;2&#3627408485;1 &#3627408485;
&#3627408486;
Increasing function

&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
>0
??????(&#3627408485;2)
??????(&#3627408485;1)
&#3627408485;2&#3627408485;1 &#3627408485;
&#3627408486;
Decreasing function
•for &#3627408485;
1<&#3627408485;
2, We observe &#3627408467;&#3627408485;1>&#3627408467;(&#3627408485;
2)

&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
<0
Increasing and Decreasing Functions

Critical Points
&#3627408486;
&#3627408485;
Critical Points
Slope of Tangent
at critical points
tan??????=tan0°=0

Find the critical points of the function
&#3627408467;&#3627408485;=&#3627408485;
4
−8&#3627408485;
2
.
A B
C D
−2,−16,0,0,(2,−16)
−2,−16,0,0,(4,128)
−4,128,0,0,(2,−16)
−4,128,0,0,(4,128)
&#3627408467;

&#3627408485;=0
⇒4&#3627408485;
3
−16&#3627408485;=0
⇒4&#3627408485;&#3627408485;
2
−4=0
⇒4&#3627408485;&#3627408485;−2(&#3627408485;+2)=0
⇒&#3627408485;=0,2,−2
&#3627408467;−2=−16
&#3627408467;0=0
&#3627408467;2=−16
So, critical points are:
−2,−16,0,0,(2,−16)

At Maxima

&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=0

&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
<0
At Minima

&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=0

&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
>0
Critical Points

Find the maxima and minima
for function &#3627408486;=&#3627408485;
3
−3&#3627408485;+2
1.Put
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=0
&#3627408465;&#3627408486;
&#3627408465;&#3627408485;
=3&#3627408485;
2
−3=0⇒&#3627408485;=+1&&#3627408485;=−1
&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=6&#3627408485;
For &#3627408485;=+1
&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=+6>0
For &#3627408485;=−1
&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
=−6<0
⇒Minima
⇒Maxima
2.Find
&#3627408465;
2
&#3627408486;
&#3627408465;&#3627408485;
2
for each value of &#3627408485;

A ball is thrown into air. Its height at any instant of
time&#3627408481;is given by ℎ=3+14&#3627408481;−5&#3627408481;
2
. What is the
maximum height attained by the ball?( &#3627408481;is in second)
Given : height ℎat time &#3627408481;, ℎ=3+14&#3627408481;−5&#3627408481;
2
Say at some time &#3627408481;the ball achieves maximum height,

&#3627408465;ℎ
&#3627408465;&#3627408481;
=14−10&#3627408481;=0⇒&#3627408481;=7/5&#3627408480;&#3627408466;&#3627408464;&#3627408476;&#3627408475;&#3627408465;
Second derivative of ℎ,
&#3627408465;
2

&#3627408465;&#3627408481;
2
=−10, −&#3627408483;&#3627408466;
Hence, at &#3627408481;=7/5&#3627408480;&#3627408466;&#3627408464;&#3627408476;&#3627408475;&#3627408465;height will be maximum

&#3627408474;&#3627408462;&#3627408485;=3+14
7
5
−5
7
5
2
=&#3627409359;&#3627409360;.&#3627409366;??????&#3627408527;&#3627408522;&#3627408533;

Integration
MathematicallyIntegrationand
differentiation are inverseof
each other.

&#3627408465;
&#3627408465;&#3627408485;
&#3627408464;=0

&#3627408465;
&#3627408465;&#3627408485;
[&#3627408472;&#3627408485;]=&#3627408472;

&#3627408465;
&#3627408465;&#3627408485;
sin&#3627408485;=cos&#3627408485;

&#3627408465;
&#3627408465;&#3627408485;
cos&#3627408485;=−sin&#3627408485;

&#3627408465;
&#3627408465;&#3627408485;
tan&#3627408485;=sec
2
&#3627408485;
•׬0&#3627408465;&#3627408485;=&#3627408464;
•׬&#3627408472;&#3627408465;&#3627408485;=&#3627408472;&#3627408485;+&#3627408464;
•׬cos&#3627408485;&#3627408465;&#3627408485;=sin&#3627408485;+&#3627408464;
•׬−sin&#3627408485;&#3627408465;&#3627408485;=cos&#3627408485;+&#3627408464;
•׬sec
2
&#3627408485;&#3627408465;&#3627408485;=tan&#3627408485;+&#3627408464;

Integrationofpolynomialfunction
Properties of Integration
•׬&#3627408485;
&#3627408475;
&#3627408465;&#3627408485;=
&#3627408485;
??????+1
&#3627408475;+1
+&#3627408464;&#3627408475;≠−1
•׬&#3627408472;&#3627408485;
&#3627408475;
&#3627408465;&#3627408485;=&#3627408472;
&#3627408485;
??????+1
&#3627408475;+1
+&#3627408464;&#3627408475;≠−1
•׬
1
&#3627408485;
&#3627408465;&#3627408485;=ln&#3627408485;+&#3627408464;
•׬&#3627408462;&#3627408467;(&#3627408485;)&#3627408465;&#3627408485;=&#3627408462;׬&#3627408467;(&#3627408485;)&#3627408465;&#3627408485;
•׬&#3627408467;&#3627408485;±&#3627408468;&#3627408485;&#3627408465;&#3627408485;=׬&#3627408467;(&#3627408485;)&#3627408465;&#3627408485;±׬&#3627408468;&#3627408485;&#3627408465;&#3627408485;
&#3627408462;&#3627408467;
1&#3627408485;=&#3627408485;
6
⇒??????
1=න&#3627408467;
1(&#3627408485;)&#3627408465;&#3627408485;=
&#3627408485;
7
7
+&#3627408464;
&#3627408463;&#3627408467;
2&#3627408485;=−10&#3627408485;
3
⇒??????
2=න&#3627408467;
2(&#3627408485;)&#3627408465;&#3627408485;=−
10&#3627408485;
4
4
+&#3627408464;
&#3627408464;&#3627408467;
3&#3627408485;=
1
&#3627408485;
⇒??????
3=න&#3627408467;
3(&#3627408485;)&#3627408465;&#3627408485;=ln(&#3627408485;)+&#3627408464;
&#3627408465;&#3627408467;
4&#3627408485;=5 ⇒??????
4=න&#3627408467;
4(&#3627408485;)&#3627408465;&#3627408485;=5&#3627408485;+&#3627408464;

න&#3627408485;
3
&#3627408465;&#3627408485;=?
A
B
C
2
3
&#3627408485;
5
+&#3627408438;
2
5
&#3627408485;
5
+&#3627408438; D
2
3
&#3627408485;
3
+&#3627408438;
2
5
&#3627408485;
3
+&#3627408438;
න&#3627408485;
3
&#3627408465;&#3627408485;=
&#3627408485;
3
2
+1
3
2
+1
+&#3627408464;=
2
5
&#3627408485;
5
+&#3627408438;

Summary
&#3627408465;
&#3627408465;&#3627408485;
&#3627408466;
&#3627408485;
=&#3627408466;
&#3627408485;
න&#3627408466;
&#3627408485;
&#3627408465;&#3627408485;=&#3627408466;
&#3627408485;
+&#3627408464;
&#3627408465;
&#3627408465;&#3627408485;
sin&#3627408485;=cos&#3627408485;නcos&#3627408485;&#3627408465;&#3627408485;=sin&#3627408485;+&#3627408464;
&#3627408465;
&#3627408465;&#3627408485;
cos&#3627408485;=−sin&#3627408485;නsin&#3627408485;&#3627408465;&#3627408485;=−cos&#3627408485;+&#3627408464;
&#3627408465;
&#3627408465;&#3627408485;
tan&#3627408485;=sec
2
&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
cosec&#3627408485;=−cosec&#3627408485;cot&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
sec&#3627408485;=sec&#3627408485;tan&#3627408485;
&#3627408465;
&#3627408465;&#3627408485;
cot&#3627408485;=−cosec
2
&#3627408485;
න−cosec&#3627408485;cot&#3627408485;&#3627408465;&#3627408485;=cosec&#3627408485;+&#3627408464;
නsec&#3627408485;tan&#3627408485;&#3627408465;&#3627408485;=sec&#3627408485;+&#3627408464;
න−cosec
2
&#3627408485;&#3627408465;&#3627408485;=cot&#3627408485;+&#3627408464;
නsec
2
&#3627408485;&#3627408465;&#3627408485;=tan&#3627408485;+&#3627408464;

Summary
If??????=׬&#3627408467;&#3627408462;&#3627408485;+&#3627408463;&#3627408465;&#3627408485;,then assume ??????=&#3627408462;&#3627408485;+&#3627408463;
and integrate further.
⇒??????
=
1
&#3627408462;
????????????
Now??????(??????)=׬&#3627408467;??????&#3627408465;&#3627408485;,then
Replace??????with&#3627408462;&#3627408485;+&#3627408463;
??????=
1
&#3627408462;
??????(&#3627408462;&#3627408485;+&#3627408463;)
??????
1=න2&#3627408485;+3
2
&#3627408465;&#3627408485;=
(&#3627409360;&#3627408537;+&#3627409361;)
3
6
+c
??????
2=නsin2&#3627408485;+3&#3627408465;&#3627408485;=
−cos&#3627409360;&#3627408537;+&#3627409361;
2
+c
??????
3=න
1
2&#3627408485;+3
&#3627408465;&#3627408485;=
log
&#3627408466;2&#3627408485;+3
2
+&#3627408464;

Integrate the function &#3627408467;&#3627408485;=&#3627408485;
6
−10&#3627408485;
3
+
1
&#3627408485;
+2sin2&#3627408485;
A
B
C
4&#3627408485;
6
−70&#3627408485;
3
+28(log
&#3627408466;&#3627408485;−cos2&#3627408485;
4
+&#3627408464;
4&#3627408485;
7
−70&#3627408485;
4
+28(log
&#3627408466;&#3627408485;−cos2&#3627408485;
28
+&#3627408464;
D
4&#3627408485;
7
−70&#3627408485;
4
+28(log
&#3627408466;&#3627408485;−cos2&#3627408485;
24
+&#3627408464;
4&#3627408485;
7
−7&#3627408485;
4
+28(log
&#3627408466;&#3627408485;−cos2&#3627408485;
24
+&#3627408464;

Summary
??????=න
2&#3627408485;
&#3627408485;
2
+1
&#3627408465;&#3627408485;
Let&#3627408481;=&#3627408485;
2
+1

&#3627408465;&#3627408481;
&#3627408465;&#3627408485;
=2&#3627408485;
⇒&#3627408465;&#3627408481;=2&#3627408485;&#3627408465;&#3627408485;
??????=න
&#3627408465;&#3627408481;
&#3627408481;
=log
&#3627408466;&#3627408481;+&#3627408464;
⇒??????=log
&#3627408466;(&#3627408485;
2
+1)+&#3627408464;
??????=න
sin&#3627408485;
cos&#3627408485;
&#3627408465;&#3627408485;
Let&#3627408481;=cos&#3627408485;

&#3627408465;&#3627408481;
&#3627408465;&#3627408485;
=−sin&#3627408485;
⇒−&#3627408465;&#3627408481;=sin&#3627408485;&#3627408465;&#3627408485;
??????=−න
&#3627408465;&#3627408481;
&#3627408481;
=−2&#3627408481;+&#3627408464;
⇒??????=−2cos&#3627408485;+&#3627408464;

??????=න&#3627408485;
2
sin&#3627408485;
3
&#3627408465;&#3627408485;=?
&#3627408481;=&#3627408485;
3

&#3627408465;&#3627408481;
&#3627408465;&#3627408485;
=3&#3627408485;
2
Let

1
3
&#3627408465;&#3627408481;=&#3627408485;
2
&#3627408465;&#3627408485;
??????=නsin&#3627408481;.
1
3
&#3627408465;&#3627408481;
??????=−
1
3
cos&#3627408481;+&#3627408464;
??????=−
1
3
cos&#3627408485;
3
+&#3627408464;

Integration
&#3627408463;∶Upper limit of Integration
&#3627408462;∶Lower limit of Integration

Find the value of
A
B
C
D
ln2+&#3627408438;
ln2
ln2+1+&#3627408438;
ln2+1

1
2
1
&#3627408485;
&#3627408465;&#3627408485;

1
2
1
&#3627408485;
&#3627408465;&#3627408485;=[ln&#3627408485;]
1
2
=ln(2)

•Divide the whole area under the curve into
infinitely small strips of width &#3627408465;&#3627408485;.We take a
strip of width &#3627408465;&#3627408485;at &#3627408485;=&#3627408485;.
•Area of shown small strip is
&#3627408465;&#3627408436;=&#3627408467;(&#3627408485;)&#3627408465;&#3627408485;
•Total area between the curve and &#3627408485;–
axis.
&#3627408436;=න
&#3627408462;
&#3627408463;
&#3627408467;(&#3627408485;)&#3627408465;&#3627408485;
Geometrical Meaning of Integration
&#3627408467;&#3627408485;
&#3627408485;
&#3627408465;&#3627408485;
&#3627408462;&#3627408463;

&#3627408436;&#3627408479;&#3627408466;&#3627408462;&#3627408476;&#3627408467;??????&#3627408479;&#3627408462;&#3627408477;&#3627408466;&#3627408487;&#3627408470;&#3627408482;&#3627408474;=
ℎ(&#3627408462;+&#3627408463;)
2
=
4(2+6)
2
=16
&#3627408436;&#3627408479;&#3627408466;&#3627408462;&#3627408476;&#3627408467;??????&#3627408479;&#3627408462;&#3627408477;&#3627408466;&#3627408487;&#3627408470;&#3627408482;&#3627408474;=න
2
6
&#3627408485;&#3627408465;&#3627408485;
=
&#3627408485;
2
2
=16
6
2
Find the area bounded by the lines
&#3627408486;=&#3627408485;,&#3627408486;=0,&#3627408485;=2and &#3627408485;=6
&#3627408486;=&#3627408485;

1)න
&#3627408462;
&#3627408464;
&#3627408467;&#3627408485;&#3627408465;&#3627408485;=න
&#3627408462;
&#3627408463;
&#3627408467;&#3627408485;&#3627408465;&#3627408485;+න
&#3627408463;
&#3627408464;
&#3627408467;&#3627408485;&#3627408465;&#3627408485;
Propertiesof definite Integration
2)න
&#3627408462;
&#3627408462;
&#3627408467;&#3627408485;&#3627408465;&#3627408485;=0
3)න
&#3627408462;
&#3627408463;
&#3627408467;&#3627408485;&#3627408465;&#3627408485;=−න
&#3627408463;
&#3627408462;
&#3627408467;&#3627408485;&#3627408465;&#3627408485;
&#3627408467;&#3627408485;
&#3627408485;
&#3627408486;
&#3627408462;&#3627408463;&#3627408464;

Evaluate the integral:න

−??????
2

??????
2
cos&#3627408485;&#3627408465;&#3627408485;
A 2 B
4
C 0
D 3

1
4
1
2&#3627408485;
2
&#3627408465;&#3627408485;+න
4
1
1
2&#3627408485;
2
&#3627408465;&#3627408485;=?
A 0 B
4
C
2 D 3

Vector
A physical quantity is a vector when,
•It has a magnitude and a direction
•It obeys laws of vector algebra

Vector Representation
Vector &#3627408488;
Magnitude Direction
&#3627408436;or Ԧ&#3627408436; መ&#3627408436;

Vector divided by its own magnitude is a vector with unit magnitude
and in direction along the parent vector.
መ&#3627408436;
&#3627408436;
Unit Vector
&#3627408436;
|&#3627408436;|
=መ&#3627408436;
Ԧ&#3627408436;=|&#3627408436;|መ&#3627408436;

Ifacarhasavelocityof20m/sintheeastdirectionasshownin&#3627408436;?
(b)Howtorepresentvelocityvectorofagirlrunningwith5m/sinopposite
direction
&#3627408437;=−05መ&#3627408436;&#3627408474;/&#3627408480;
&#3627408436;=20መ&#3627408436;&#3627408474;/&#3627408480;
If the car’s velocity of 20 m/s along east is represented by Ԧ&#3627408436;, then how
can we represent velocity of a girl moving at 5&#3627408474;/&#3627408480;in the opposite
direction?
If the car’s velocity of 20&#3627408474;/&#3627408480;along east is represented by Ԧ&#3627408436;, then how can
we represent velocity of a girl moving at 5&#3627408474;/&#3627408480;in the opposite direction?

Cartesian Co-ordinate System
Representation of Unit Vector along Coordinate Axes

•A vector can be displaced parallel to itself as it
does not change its magnitude and direction.
•If a vector is rotated through an angle other
than multiple of 2??????(or360°) the vector
changes.
Properties of a Vector
Angle between Vectors
Step 1:
Place vectors tail to tail or head to head.
Step 2:
Measure the smaller angle i.e.angle
which is less than 180°between them.

(i)&#3627408436;and &#3627408437;is 60°
(ii) &#3627408436;and &#3627408438;is 45°
(iii) &#3627408437;and &#3627408438;is 105°
Three Vectors are arranged along sides of triangle as
shown. Find the angle between the vectors.
(i)
(ii)
(iii)

Multiplication of a Vector with a Scalar

Equal Vectors
•Ԧ&#3627408436;=&#3627408437;=Ԧ&#3627408438;Equal magnitude
•መ&#3627408436;=෠&#3627408437;=መ&#3627408438;Same direction
Parallel vectors
Anglebetween anytwo
vectorsis0°or180°thenthey
arealongthesamedirection
orexactlyoppositedirections.
•??????=0°or 180°
•Ԧ&#3627408436;≠&#3627408437;
•Ԧ&#3627408436;∥&#3627408437; Null vector
•Ԧ&#3627408436;=0
Iftwovectorsareco-linearor
parallelthenonecanbe
expressedinthetermsof
anotheras,
•Ԧ&#3627408462;=??????&#3627408463;(where??????isconstant)
Collinear vectors
Types of Vectors

Three(ormore)vectorsare
calledcoplanarvectorsifthey
lieinthesameplane.Two(free)
vectorscanbemadecoplanar.
Coplanar vectors
Same in magnitude but opposite in
direction.
•መ&#3627408438;=−෡&#3627408439;
•Ԧ&#3627408438;=&#3627408439;
Negative of a vectors
Types of Vectors

Magnitude of a Vector and
Displacement Vector
Displacement Vector is the change in
position vector.
Ԧ&#3627408480;=&#3627408485;
2−&#3627408485;
1Ƹ&#3627408470;+&#3627408486;
2−&#3627408486;
1Ƹ&#3627408471;+(&#3627408487;
2−&#3627408487;
1)෠&#3627408472;
Ԧ&#3627408479;=&#3627408462;Ƹ&#3627408470;+&#3627408463;Ƹ&#3627408471;
Ԧ&#3627408479;=Length of line &#3627408450;&#3627408436;
Ԧ&#3627408479;=&#3627408462;
2
+&#3627408463;
2

Position vector gives us the position of a point in magnitude and direction.
Ԧ&#3627408479;
1=&#3627408462;Ƹ&#3627408470;+&#3627408463;Ƹ&#3627408471;
Position Vector in 2D and 3D
Position vector
Ԧ&#3627408479;
1=&#3627408485;
1Ƹ&#3627408470;+&#3627408486;
1Ƹ&#3627408471;+&#3627408487;
1
෠&#3627408472;
Ԧ&#3627408479;
2=&#3627408485;
2Ƹ&#3627408470;+&#3627408486;
2Ƹ&#3627408471;+&#3627408487;
2
෠&#3627408472;
Magnitude
Ԧ&#3627408479;
1=&#3627408485;
1
2
+&#3627408486;
1
2
+&#3627408487;
1
2
Ԧ&#3627408479;
2=&#3627408485;
2
2
+&#3627408486;
2
2
+&#3627408487;
2
2
Ԧ&#3627408479;
1=&#3627408462;
2
+&#3627408463;
2

Magnitude
&#3627408453;=&#3627408453;
&#3627408485;Ƹ&#3627408470;+&#3627408453;
&#3627408486;Ƹ&#3627408471;
Direction
&#3627408453;=&#3627408453;
&#3627408485;
2
+&#3627408453;
&#3627408486;
2
tan??????=
&#3627408453;
&#3627408486;
&#3627408453;
&#3627408485;
&#3627408486;
&#3627408485;
&#3627408453;
&#3627408450;
??????
Magnitude and Direction
&#3627408453;=&#3627408453;
&#3627408485;+&#3627408453;
&#3627408486;
&#3627408453;=&#3627408453;
&#3627408485;Ƹ&#3627408470;+&#3627408453;
&#3627408486;Ƹ&#3627408471;
&#3627408453;=&#3627408453;cos??????Ƹ&#3627408470;+&#3627408453;sin??????Ƹ&#3627408471;

&#3627408486;
&#3627408485;
&#3627408436;=30
&#3627408450;
&#3627408486;
&#3627408437;=80
&#3627408450;
Fill in the Blanks
Ԧ&#3627408436;= Ƹ&#3627408470;+ Ƹ&#3627408471;−152 152 &#3627408437;= Ƹ&#3627408470;+ Ƹ&#3627408471;−40 −403
&#3627408485;

Two friends &#3627408436;and &#3627408437;are running parallel to each other.
Velocity of &#3627408436;is 3Ƹ&#3627408470;+4Ƹ&#3627408471;&#3627408474;/&#3627408480;and speed of &#3627408437;is 20&#3627408474;/&#3627408480;. Find
the velocity of &#3627408437;?
&#3627408437;=&#3627408437;෠&#3627408437;
&#3627408437;=20×
3Ƹ&#3627408470;+4Ƹ&#3627408471;
5
Ԧ&#3627408436;=3
2
+4
2
=5
Unit vector along Ԧ&#3627408436;,
መ&#3627408436;=
3Ƹ&#3627408470;+4Ƹ&#3627408471;
5
&#3627408437;=12Ƹ&#3627408470;+16Ƹ&#3627408471;
መ&#3627408436;=෠&#3627408437;
መ&#3627408436;=
Ԧ&#3627408436;
Ԧ&#3627408436;

Steps for vector Addition
&#3627408486;
&#3627408485;
Ԧ&#3627408451;
??????

&#3627408451;
&#3627408485;
&#3627408452;
&#3627408486;
&#3627408452;
&#3627408485;
&#3627408451;
&#3627408486;
&#3627408452;
Ԧ&#3627408451;=&#3627408451;
&#3627408485;Ƹ&#3627408470;+&#3627408451;
&#3627408486;Ƹ&#3627408471;
&#3627408452;=&#3627408452;
&#3627408485;Ƹ&#3627408470;+&#3627408452;&#3627408486;Ƹ&#3627408471;
Ԧ&#3627408451;=&#3627408451;cos??????Ƹ&#3627408470;+&#3627408451;sin??????Ƹ&#3627408471;
&#3627408452;=&#3627408452;cos??????Ƹ&#3627408470;+&#3627408452;sin??????Ƹ&#3627408471;
Ԧ&#3627408451;+&#3627408452;=&#3627408451;
&#3627408485;+&#3627408452;
&#3627408485;Ƹ&#3627408470;+&#3627408451;
&#3627408486;+&#3627408452;
&#3627408486;Ƹ&#3627408471;
Ԧ&#3627408451;+&#3627408452;=&#3627408451;cos??????+&#3627408452;cos??????Ƹ&#3627408470;+&#3627408451;sin??????+&#3627408452;sin??????Ƹ&#3627408471;

Ԧ&#3627408451;+&#3627408452;=
9
2
Ƹ&#3627408471;+
93
2
Ƹ&#3627408470;+3Ƹ&#3627408471;−33Ƹ&#3627408470;
Ԧ&#3627408451;+&#3627408452;=
9
2
Ƹ&#3627408471;+3Ƹ&#3627408471;+
93
2
Ƹ&#3627408470;−33Ƹ&#3627408470;
Ԧ&#3627408451;+&#3627408452;=
15
2
Ƹ&#3627408471;+
33
2
Ƹ&#3627408470;
30°
&#3627408486;
&#3627408485;
6
30°
3
33
9
93
2
9
2
IfԦ&#3627408451;=9unitsand &#3627408452;=6units, thenfindԦ&#3627408451;+&#3627408452;.

Laws of Vector
Addition/Subtraction
Laws of Vector Addition/Subtraction
Triangle Law
Parallelogram
Law
Polygon Law

&#3627408453;=&#3627408436;
2
+&#3627408437;
2
+2&#3627408436;&#3627408437;cos??????
&#3627408486;
??????
&#3627408437;
&#3627408453;
&#3627408436;Ƹ&#3627408470; &#3627408485;
Ԧ&#3627408436;
&#3627408437;sin??????Ƹ&#3627408471;
&#3627408453;
&#3627408437;cos??????Ƹ&#3627408470;
&#3627408453;=Ԧ&#3627408436;+&#3627408437;=&#3627408436;+&#3627408437;cos??????Ƹ&#3627408470;+&#3627408437;sin??????Ƹ&#3627408471;
&#3627408453;=Ԧ&#3627408436;+&#3627408437;
&#3627408453;=&#3627408436;+&#3627408437;cos??????
2
+&#3627408437;sin??????
2
&#3627408453;=&#3627408436;
2
+&#3627408437;
2
cos
2
??????+2&#3627408436;&#3627408437;cos??????+&#3627408437;
2
sin
2
??????
Triangle Law

When two vectors with common origin
represent two adjacent sides of a
parallelogramin magnitude and direction,
then the resultant vector is represented
both in magnitude and direction by the
diagonal passing through that point.
&#3627408486;
&#3627408453;=&#3627408436;
2
+&#3627408437;
2
+2&#3627408436;&#3627408437;cos??????
Here &#3627409148;is the angle between Ԧ&#3627408436;and &#3627408453;
tan&#3627409148;=
&#3627408437;sin??????
Ԧ&#3627408436;+&#3627408437;cos??????
Parallelogram Law

&#3627408453;
3
&#3627408439;
Ԧ&#3627408436;
&#3627408437;
Ԧ&#3627408438;
•Join vectors in same order(head to tail).
•Resultant Vector –closing side of polygon
in opposite order
&#3627408453;
3=Ԧ&#3627408436;+&#3627408437;+Ԧ&#3627408438;+&#3627408439;
Polygon Law of Vector Addition

•Ԧ&#3627408436;+&#3627408437;: Ԧ&#3627408436;and&#3627408437;must be same physical quantities.
•Commutative Property :Ԧ&#3627408436;+&#3627408437;= &#3627408437;+Ԧ&#3627408436;
•Associative Property :(Ԧ&#3627408436;+&#3627408437;)+Ԧ&#3627408438;=Ԧ&#3627408436;+(&#3627408437;+Ԧ&#3627408438;)
Properties of Vector Addition

??????
&#3627408437;
&#3627408436;
Ԧ&#3627408436;.&#3627408437;=&#3627408436;&#3627408437;cos??????
The dot product of two vectors &#3627408436;and &#3627408437;is
equal to the products of their magnitude
times the cosine of angle between them.
Scalar / Dot Product
Ԧ&#3627408436;.&#3627408437;=Ԧ&#3627408436;&#3627408437;cos??????
&#3627408436;=&#3627408436;
&#3627408485;Ƹ&#3627408470;+&#3627408436;
&#3627408486;Ƹ&#3627408471;+&#3627408436;
&#3627408487;
෠&#3627408472;,
&#3627408437;=&#3627408437;
&#3627408485;Ƹ&#3627408470;+&#3627408437;
&#3627408486;Ƹ&#3627408471;+&#3627408437;
&#3627408487;
෠&#3627408472;
&#3627408436;.&#3627408437;=&#3627408436;
&#3627408485;&#3627408437;
&#3627408485;+&#3627408436;
&#3627408486;&#3627408437;
&#3627408486;+&#3627408436;
&#3627408487;&#3627408437;
&#3627408487;

60
°
&#3627408437;
&#3627408436;
If&#3627408436;=2,&#3627408437;=3and the angle between
&#3627408436;and&#3627408437;is60°. Find&#3627408436;.&#3627408437;
&#3627408436;.&#3627408437;=|&#3627408436;||&#3627408437;|cos??????
=2×3×cos60
°
=2×3×
1
2
=3
Ԧ&#3627408436;.&#3627408437;=3

•Dot product of two vectors is always (commutative), i.e. &#3627408436;.&#3627408437;=&#3627408437;.&#3627408436;
•Dot product of two vectors is always distributive, i.e. &#3627408436;.&#3627408437;+&#3627408438;=&#3627408436;.&#3627408437;+&#3627408436;.&#3627408438;
•Dot product of two vectors will be maximumwhen they are parallel(i.e., angle
between them is zero).
Properties of Dot Product
&#3627408436;.&#3627408437;is positive
??????<&#3627409367;&#3627409358;
°
⇒????????????????????????>&#3627409358;
Ԧ&#3627408436;.&#3627408437;=&#3627408436;&#3627408437;cos??????
??????=&#3627409367;&#3627409358;
°
&#3627408436;.&#3627408437;is zero
&#3627408436;.&#3627408437;is negative
??????>&#3627409367;&#3627409358;
°
⇒????????????????????????<&#3627409358;
•&#3627408436;.&#3627408436;=&#3627408436;
2 •ො&#3627408475;.ො&#3627408475;=1 •Ƹ&#3627408470;.Ƹ&#3627408471;=Ƹ&#3627408471;.෠&#3627408472;=෠&#3627408472;.Ƹ&#3627408470;=0

&#3627408436;.&#3627408437;=0
⟹&#3627408436;&#3627408437;cos??????=0
⟹cos??????=0
??????=cos
−1
0=90
°
&#3627408436;
&#3627408437;
90
°
Thus, &#3627408436;.&#3627408437;=0is a condition fortwo
vectors to be orthogonal.
If the scalar product of two non-zero vectors
becomes zero. What is the angle between them?

Thecrossproductoftwovectors
&#3627408436;and&#3627408437;isequaltotheproducts
oftheirmagnitudetimesthe
sineofanglebetweenthemand
directionperpendiculartothe
planecontainingthetwo
vectors.
Ԧ&#3627408436;×&#3627408437;=Ԧ&#3627408436;&#3627408437;sin??????ො&#3627408475;
Ԧ&#3627408436;×&#3627408437;=Ԧ&#3627408436;&#3627408437;sin??????
Vector / Cross Product
Here, ො&#3627408475;is a unit vector
perpendicularto both Ԧ&#3627408436;and &#3627408437;.
Ԧ&#3627408436;×&#3627408437;=Ԧ&#3627408436;&#3627408437;sin??????ො&#3627408475;

To find the direction of Ԧ&#3627408436;×&#3627408437;:
•Draw Ԧ&#3627408436;and &#3627408437;tail-to-tail.
•Place the stretched right palm such that the fingers are along Ԧ&#3627408436;and when the
fingers are closed, they go towards &#3627408437;.
•The direction in which thumb points gives the direction of Ԧ&#3627408438;=Ԧ&#3627408436;×&#3627408437;.
Right Hand Thumb Rule
Ԧ&#3627408436;
&#3627408437;Ԧ&#3627408436;&#3627408437;
Ԧ&#3627408438;

Ԧ&#3627408436;×&#3627408437;=Ԧ&#3627408436;&#3627408437;sin??????
=5×4×sin30°
=10
Using the right-hand thumb rule, the
direction of Ԧ&#3627408436;×&#3627408437;is along −&#3627408487;axis
Find the magnitude and direction of Ԧ&#3627408436;×&#3627408437;and
&#3627408437;×Ԧ&#3627408436;, where Ԧ&#3627408436;&&#3627408437;lie in &#3627408485;−&#3627408486;planeas shown.
The direction of &#3627408437;×Ԧ&#3627408436;is along +&#3627408487;axis.
⟹Ԧ&#3627408436;×&#3627408437;=−10෠&#3627408472;
⟹&#3627408437;×Ԧ&#3627408436;=10෠&#3627408472;
Ԧ&#3627408436;
&#3627408437;
30°
4
5

In the right-handed coordinate system, the
coordinate axes &#3627408485;,&#3627408486;and &#3627408487;are chosen such
that bending the fingers of the right hand
from &#3627408485;to &#3627408486;will lead the thumb along the
&#3627408487;−axis.
Ƹ&#3627408470;
Ƹ&#3627408471;
෠&#3627408472;
Ƹ&#3627408470;×Ƹ&#3627408471;=෠&#3627408472;,Ƹ&#3627408471;×Ƹ&#3627408470;=−෠&#3627408472;
Ƹ&#3627408471;×෠&#3627408472;=Ƹ&#3627408470;,෠&#3627408472;×Ƹ&#3627408471;=−Ƹ&#3627408470;
෠&#3627408472;×Ƹ&#3627408470;=Ƹ&#3627408471;,Ƹ&#3627408470;×෠&#3627408472;=−Ƹ&#3627408471;
Cross Product of Orthogonal Unit Vectors

Cross Product in Component Form
Ԧ&#3627408436;×&#3627408437;=&#3627408436;
&#3627408485;Ƹ&#3627408470;+&#3627408436;
&#3627408486;Ƹ&#3627408471;+&#3627408436;
&#3627408487;
෠&#3627408472;×&#3627408437;
&#3627408485;Ƹ&#3627408470;+&#3627408437;
&#3627408486;Ƹ&#3627408471;+&#3627408437;
&#3627408487;
෠&#3627408472;
Ԧ&#3627408436;×&#3627408437;=&#3627408436;
&#3627408485;&#3627408437;
&#3627408485;Ƹ&#3627408470;×Ƹ&#3627408470;+&#3627408436;
&#3627408486;&#3627408437;
&#3627408485;Ƹ&#3627408471;×Ƹ&#3627408470;+&#3627408436;
&#3627408487;&#3627408437;
&#3627408485;
෠&#3627408472;×Ƹ&#3627408470;
+&#3627408436;
&#3627408485;&#3627408437;
&#3627408486;Ƹ&#3627408470;×Ƹ&#3627408471;+&#3627408436;
&#3627408486;&#3627408437;
&#3627408486;Ƹ&#3627408471;×Ƹ&#3627408471;+&#3627408436;
&#3627408487;&#3627408437;
&#3627408486;
෠&#3627408472;×Ƹ&#3627408471;
+&#3627408436;
&#3627408485;&#3627408437;
&#3627408487;Ƹ&#3627408470;×෠&#3627408472;+&#3627408436;
&#3627408486;&#3627408437;
&#3627408487;Ƹ&#3627408471;×෠&#3627408472;+&#3627408436;
&#3627408487;&#3627408437;
&#3627408487;
෠&#3627408472;×෠&#3627408472;
Ԧ&#3627408436;×&#3627408437;=(&#3627408436;
&#3627408486;&#3627408437;
&#3627408487;−&#3627408436;
&#3627408487;&#3627408437;
&#3627408486;)Ƹ&#3627408470;−&#3627408436;
&#3627408485;&#3627408437;
&#3627408487;−&#3627408436;
&#3627408487;&#3627408437;
&#3627408485;Ƹ&#3627408471;+(&#3627408436;
&#3627408485;&#3627408437;
&#3627408486;−&#3627408436;
&#3627408486;&#3627408437;
&#3627408485;)෠&#3627408472;
Ƹ&#3627408470;
Ƹ&#3627408471;
෠&#3627408472;
Ԧ&#3627408436;=&#3627408436;
&#3627408485;Ƹ&#3627408470;+&#3627408436;
&#3627408486;Ƹ&#3627408471;+&#3627408436;
&#3627408487;
෠&#3627408472;&#3627408437;=&#3627408437;
&#3627408485;Ƹ&#3627408470;+&#3627408437;
&#3627408486;Ƹ&#3627408471;+&#3627408437;
&#3627408487;
෠&#3627408472;

Ԧ&#3627408436;×&#3627408437;=(&#3627408436;
&#3627408486;&#3627408437;
&#3627408487;−&#3627408436;
&#3627408487;&#3627408437;
&#3627408486;)Ƹ&#3627408470;−&#3627408436;
&#3627408485;&#3627408437;
&#3627408487;−&#3627408436;
&#3627408487;&#3627408437;
&#3627408485;Ƹ&#3627408471;+(&#3627408436;
&#3627408485;&#3627408437;
&#3627408486;−&#3627408436;
&#3627408486;&#3627408437;
&#3627408485;)෠&#3627408472;
Ԧ&#3627408436;×&#3627408437;=2Ƹ&#3627408470;−5Ƹ&#3627408471;+3෠&#3627408472;×3Ƹ&#3627408470;+4Ƹ&#3627408471;−9෠&#3627408472;
=6Ƹ&#3627408470;×Ƹ&#3627408470;−15Ƹ&#3627408471;×Ƹ&#3627408470;+9෠&#3627408472;×Ƹ&#3627408470;+8Ƹ&#3627408470;×Ƹ&#3627408471;−20Ƹ&#3627408471;×Ƹ&#3627408471;
+12෠&#3627408472;×Ƹ&#3627408471;−18Ƹ&#3627408470;×෠&#3627408472;+45Ƹ&#3627408471;×෠&#3627408472;−27(෠&#3627408472;×෠&#3627408472;)
Ԧ&#3627408436;×&#3627408437;=Ƹ&#3627408470;45−12−Ƹ&#3627408471;−18−9+෠&#3627408472;8−(−15)
Ԧ&#3627408436;×&#3627408437;=33Ƹ&#3627408470;+27Ƹ&#3627408471;+23෠&#3627408472;
Find Ԧ&#3627408436;×&#3627408437;, where Ԧ&#3627408436;=2Ƹ&#3627408470;−5Ƹ&#3627408471;+3෠&#3627408472;and &#3627408437;=3Ƹ&#3627408470;+
4Ƹ&#3627408471;−9෠&#3627408472;.