HC Verma Summary: Physics and Mathematics | Physics Class 11 - NEET PDF Download

 Page 1


1 Introduction to Ph ysics and Mathematics
Ph ysics is about understanding nature’ s rules, lik e how objects move or how
forces work. Mathematics is the language that mak es these rules clear and pre-
cise. T o describe ph ysical phenomena, we use tools lik e algebr a, trigonome-
try , geometry , vector algebr a, differential calculus, and integr al calculus. This
chapter focuses on vectors, calculus, significant digits, and measurement errors,
which are essential for solving ph ysics problems. A strong gr asp of these mathe-
matical tools helps us describe, predict, and apply ph ysical principles effectively .
K ey Idea : Mathematics provides a clear and precise wa y to express ph ysics con-
cepts, making them easier to understand and apply .
2 V ectors and Scalars
Ph ysical quantities are either scalars or vectors . Scalars, lik e mass or temper a-
ture, are described b y a number and a unit (e.g., 6 k g). They add using regular
arithmetic (e.g., 6 k g + 3 k g = 9 k g). V ectors, lik e velocity or force, need both a mag-
nitude and a direction (e.g., 5 m/s north). V ectors are represented with an arrow
(e.g.,? v ) or bold letters (e.g., v ) and are dr awn as lines with an arrow indicating
direction, where the length represents magnitude.
V ectors add using the triangle rule : dr aw the first vector , place the tail of the
second vector at the head of the first, and the resultant is the vector from the
tail of the first to the head of the second. F or example, if a particle moves 3 m/s
east and 4 m/s north, the resultant velocity is found b y dr awing a r ight triangle,
yielding a magnitude of
v
3
2
+4
2
= 5 m/s at an angle tan
-1
(4/3) ˜ 53
?
north of
east.
1
Page 2


1 Introduction to Ph ysics and Mathematics
Ph ysics is about understanding nature’ s rules, lik e how objects move or how
forces work. Mathematics is the language that mak es these rules clear and pre-
cise. T o describe ph ysical phenomena, we use tools lik e algebr a, trigonome-
try , geometry , vector algebr a, differential calculus, and integr al calculus. This
chapter focuses on vectors, calculus, significant digits, and measurement errors,
which are essential for solving ph ysics problems. A strong gr asp of these mathe-
matical tools helps us describe, predict, and apply ph ysical principles effectively .
K ey Idea : Mathematics provides a clear and precise wa y to express ph ysics con-
cepts, making them easier to understand and apply .
2 V ectors and Scalars
Ph ysical quantities are either scalars or vectors . Scalars, lik e mass or temper a-
ture, are described b y a number and a unit (e.g., 6 k g). They add using regular
arithmetic (e.g., 6 k g + 3 k g = 9 k g). V ectors, lik e velocity or force, need both a mag-
nitude and a direction (e.g., 5 m/s north). V ectors are represented with an arrow
(e.g.,? v ) or bold letters (e.g., v ) and are dr awn as lines with an arrow indicating
direction, where the length represents magnitude.
V ectors add using the triangle rule : dr aw the first vector , place the tail of the
second vector at the head of the first, and the resultant is the vector from the
tail of the first to the head of the second. F or example, if a particle moves 3 m/s
east and 4 m/s north, the resultant velocity is found b y dr awing a r ight triangle,
yielding a magnitude of
v
3
2
+4
2
= 5 m/s at an angle tan
-1
(4/3) ˜ 53
?
north of
east.
1
3 m/ s
Quantities with magnitude and direction but not following the triangle rule (e.g.,
electric current) are not vectors.
K ey Idea : Scalars have only magnitude; vectors have magnitude and direction
and add via the triangle rule.
2.1 Solved Example: V ector A ddition
Question : A cyclist rides at 12 m/ s east, and a wind pushes them at 9 m/ s north.
Find t he resultant velocity’ s magnitude and direction.
Solution : The velocities are perpendicular (? = 90
?
).
|? v| =
v
12
2
+9
2
=
v
144+81 =
v
225 = 15 m/ s
Angle = tan
-1
(
9
12
)
= tan
-1
(0.75)˜ 36.87
?
north of east
The resultant velocity is 15 m/ s at36.87
?
north of east.
2.2 W ork ed-Out Example 1 (A dapted)
Question : A vector has an x-component of 20 units and a y-component of 48
units. Find its magnitude and direction.
Solution : The vector is the resultant of two perpendicular components: 20 units
along x-axis, 48 units along y-axis.
Magnitude =
v
20
2
+48
2
=
v
400+2304 =
v
2704 = 52 units
Angle with x-axis = tan
-1
(
48
20
)
= tan
-1
(2.4)˜ 67.38
?
The vector has magnitude 52 units at 67.38
?
from the x-axis.
3 Equality of V ectors
Two vectors are equal if they have the same magnitude and direction. F or exam-
ple, a velocity of 10 m/s west at one point is equal to 10 m/s west at another point,
2
Page 3


1 Introduction to Ph ysics and Mathematics
Ph ysics is about understanding nature’ s rules, lik e how objects move or how
forces work. Mathematics is the language that mak es these rules clear and pre-
cise. T o describe ph ysical phenomena, we use tools lik e algebr a, trigonome-
try , geometry , vector algebr a, differential calculus, and integr al calculus. This
chapter focuses on vectors, calculus, significant digits, and measurement errors,
which are essential for solving ph ysics problems. A strong gr asp of these mathe-
matical tools helps us describe, predict, and apply ph ysical principles effectively .
K ey Idea : Mathematics provides a clear and precise wa y to express ph ysics con-
cepts, making them easier to understand and apply .
2 V ectors and Scalars
Ph ysical quantities are either scalars or vectors . Scalars, lik e mass or temper a-
ture, are described b y a number and a unit (e.g., 6 k g). They add using regular
arithmetic (e.g., 6 k g + 3 k g = 9 k g). V ectors, lik e velocity or force, need both a mag-
nitude and a direction (e.g., 5 m/s north). V ectors are represented with an arrow
(e.g.,? v ) or bold letters (e.g., v ) and are dr awn as lines with an arrow indicating
direction, where the length represents magnitude.
V ectors add using the triangle rule : dr aw the first vector , place the tail of the
second vector at the head of the first, and the resultant is the vector from the
tail of the first to the head of the second. F or example, if a particle moves 3 m/s
east and 4 m/s north, the resultant velocity is found b y dr awing a r ight triangle,
yielding a magnitude of
v
3
2
+4
2
= 5 m/s at an angle tan
-1
(4/3) ˜ 53
?
north of
east.
1
3 m/ s
Quantities with magnitude and direction but not following the triangle rule (e.g.,
electric current) are not vectors.
K ey Idea : Scalars have only magnitude; vectors have magnitude and direction
and add via the triangle rule.
2.1 Solved Example: V ector A ddition
Question : A cyclist rides at 12 m/ s east, and a wind pushes them at 9 m/ s north.
Find t he resultant velocity’ s magnitude and direction.
Solution : The velocities are perpendicular (? = 90
?
).
|? v| =
v
12
2
+9
2
=
v
144+81 =
v
225 = 15 m/ s
Angle = tan
-1
(
9
12
)
= tan
-1
(0.75)˜ 36.87
?
north of east
The resultant velocity is 15 m/ s at36.87
?
north of east.
2.2 W ork ed-Out Example 1 (A dapted)
Question : A vector has an x-component of 20 units and a y-component of 48
units. Find its magnitude and direction.
Solution : The vector is the resultant of two perpendicular components: 20 units
along x-axis, 48 units along y-axis.
Magnitude =
v
20
2
+48
2
=
v
400+2304 =
v
2704 = 52 units
Angle with x-axis = tan
-1
(
48
20
)
= tan
-1
(2.4)˜ 67.38
?
The vector has magnitude 52 units at 67.38
?
from the x-axis.
3 Equality of V ectors
Two vectors are equal if they have the same magnitude and direction. F or exam-
ple, a velocity of 10 m/s west at one point is equal to 10 m/s west at another point,
2
as long as magnitude and direction are unchanged. Moving a vector par allel to
itself (par allel tr anslation) does not alter it.
K ey Idea : Equal vectors have identical magnitudes and directions, regardless of
position.
3.1 Solved Example: V ector Equality
Question : Are two forces
?
F
1
= 25 N at45
?
north of east and
?
F
2
= 25 N at45
?
north
of e ast equal?
Solution : Both have the same magnitude (25 N) and direction (45
?
north of east).
Thus,
?
F
1
=
?
F
2
.
4 A ddition of V ectors
T o add vectors? a and
?
b , use the triangle rule : dr aw? a , then dr aw
?
b from the head of
? a . The vector from the tail of? a to the head of
?
b is? a+
?
b . Alternatively , the par allelo-
gr am rule dr aws both vectors from a common point, completes a par allelogr am,
and tak es the diagonal through the common tail as the sum.
F or ma gnitudesa ,b , and angle? between them, the resultant’ s m agnitude is:
|? a+
?
b| =
v
a
2
+b
2
+2ab cos?
The anglea with? a is:
tana =
b sin?
a+b cos?
? a
K ey Idea : V ector addition uses geometric rules to combine magnitude and di-
rection.
4.1 Solved Example: V ector A ddition
Question : Two forces
?
F
1
= 30 N at0
?
(x-axis) and
?
F
2
= 40 N at45
?
act on an object.
Find t he resultant force’ s magnitude and direction.
Solution :
|
?
F
1
+
?
F
2
| =
v
30
2
+40
2
+2×30×40 cos45
?
=
v
900+1600+2400×0.707˜
v
4098.8˜ 64.04 N
tana =
40 sin45
?
30+40 cos45
?
=
40×0.707
30+40×0.707
˜
28.28
58.28
˜ 0.485
3
Page 4


1 Introduction to Ph ysics and Mathematics
Ph ysics is about understanding nature’ s rules, lik e how objects move or how
forces work. Mathematics is the language that mak es these rules clear and pre-
cise. T o describe ph ysical phenomena, we use tools lik e algebr a, trigonome-
try , geometry , vector algebr a, differential calculus, and integr al calculus. This
chapter focuses on vectors, calculus, significant digits, and measurement errors,
which are essential for solving ph ysics problems. A strong gr asp of these mathe-
matical tools helps us describe, predict, and apply ph ysical principles effectively .
K ey Idea : Mathematics provides a clear and precise wa y to express ph ysics con-
cepts, making them easier to understand and apply .
2 V ectors and Scalars
Ph ysical quantities are either scalars or vectors . Scalars, lik e mass or temper a-
ture, are described b y a number and a unit (e.g., 6 k g). They add using regular
arithmetic (e.g., 6 k g + 3 k g = 9 k g). V ectors, lik e velocity or force, need both a mag-
nitude and a direction (e.g., 5 m/s north). V ectors are represented with an arrow
(e.g.,? v ) or bold letters (e.g., v ) and are dr awn as lines with an arrow indicating
direction, where the length represents magnitude.
V ectors add using the triangle rule : dr aw the first vector , place the tail of the
second vector at the head of the first, and the resultant is the vector from the
tail of the first to the head of the second. F or example, if a particle moves 3 m/s
east and 4 m/s north, the resultant velocity is found b y dr awing a r ight triangle,
yielding a magnitude of
v
3
2
+4
2
= 5 m/s at an angle tan
-1
(4/3) ˜ 53
?
north of
east.
1
3 m/ s
Quantities with magnitude and direction but not following the triangle rule (e.g.,
electric current) are not vectors.
K ey Idea : Scalars have only magnitude; vectors have magnitude and direction
and add via the triangle rule.
2.1 Solved Example: V ector A ddition
Question : A cyclist rides at 12 m/ s east, and a wind pushes them at 9 m/ s north.
Find t he resultant velocity’ s magnitude and direction.
Solution : The velocities are perpendicular (? = 90
?
).
|? v| =
v
12
2
+9
2
=
v
144+81 =
v
225 = 15 m/ s
Angle = tan
-1
(
9
12
)
= tan
-1
(0.75)˜ 36.87
?
north of east
The resultant velocity is 15 m/ s at36.87
?
north of east.
2.2 W ork ed-Out Example 1 (A dapted)
Question : A vector has an x-component of 20 units and a y-component of 48
units. Find its magnitude and direction.
Solution : The vector is the resultant of two perpendicular components: 20 units
along x-axis, 48 units along y-axis.
Magnitude =
v
20
2
+48
2
=
v
400+2304 =
v
2704 = 52 units
Angle with x-axis = tan
-1
(
48
20
)
= tan
-1
(2.4)˜ 67.38
?
The vector has magnitude 52 units at 67.38
?
from the x-axis.
3 Equality of V ectors
Two vectors are equal if they have the same magnitude and direction. F or exam-
ple, a velocity of 10 m/s west at one point is equal to 10 m/s west at another point,
2
as long as magnitude and direction are unchanged. Moving a vector par allel to
itself (par allel tr anslation) does not alter it.
K ey Idea : Equal vectors have identical magnitudes and directions, regardless of
position.
3.1 Solved Example: V ector Equality
Question : Are two forces
?
F
1
= 25 N at45
?
north of east and
?
F
2
= 25 N at45
?
north
of e ast equal?
Solution : Both have the same magnitude (25 N) and direction (45
?
north of east).
Thus,
?
F
1
=
?
F
2
.
4 A ddition of V ectors
T o add vectors? a and
?
b , use the triangle rule : dr aw? a , then dr aw
?
b from the head of
? a . The vector from the tail of? a to the head of
?
b is? a+
?
b . Alternatively , the par allelo-
gr am rule dr aws both vectors from a common point, completes a par allelogr am,
and tak es the diagonal through the common tail as the sum.
F or ma gnitudesa ,b , and angle? between them, the resultant’ s m agnitude is:
|? a+
?
b| =
v
a
2
+b
2
+2ab cos?
The anglea with? a is:
tana =
b sin?
a+b cos?
? a
K ey Idea : V ector addition uses geometric rules to combine magnitude and di-
rection.
4.1 Solved Example: V ector A ddition
Question : Two forces
?
F
1
= 30 N at0
?
(x-axis) and
?
F
2
= 40 N at45
?
act on an object.
Find t he resultant force’ s magnitude and direction.
Solution :
|
?
F
1
+
?
F
2
| =
v
30
2
+40
2
+2×30×40 cos45
?
=
v
900+1600+2400×0.707˜
v
4098.8˜ 64.04 N
tana =
40 sin45
?
30+40 cos45
?
=
40×0.707
30+40×0.707
˜
28.28
58.28
˜ 0.485
3
a˜ tan
-1
(0.485)˜ 25.9
?
The re sultant is64.04 N at25.9
?
from the x-axis.
4.2 W o rk ed-Out Example 2 (A dapted)
Question : Find the resultant of three vectors: 6 m at37
?
, 4 m along x-axis, and 3
m al ong y-axis.
Solution : Resolve into x and y components. - 6 m at 37
?
: x = 6 cos37
?
˜ 4.8 m,
y = 6 sin37
?
˜ 3.6 m. - 4 m along x-axis: x = 4 m, y = 0 . - 3 m along y-axis: x = 0 ,
y = 3 m.
T otal x-component: 4.8+4+0 = 8.8 m. T otal y-component: 3.6+0+3 = 6.6 m.
Magnitude =
v
8.8
2
+6.6
2
˜
v
77.44+43.56˜ 11 m
Angle = tan
-1
(
6.6
8.8
)
˜ 36.87
?
The r esultant is approximately 11 m at 36.87
?
.
5 M ultiplication of a V ector b y a Number
Multiplying a vector? a with magnitudea b y a numberk gives
?
b = k? a , with magni-
tude|k|·a . Ifk > 0 , the direction is unchanged; ifk < 0 , it reverses. F or example,
if? a = 4 m/ s east, then-2? a = 8 m/ s west. A vector can be written as? a = a? u , where
? u is a unit vector (magnitude 1) in? a ’ s direction.
K ey Idea : Scalar multiplication scales a vector ’ s magnitude and ma y reverse its
direction.
5.1 Solved Example: Scalar Multiplication
Question : A velocity? v = 15 m/ s south is multiplied b y 2.5. Find the new vector ’ s
magnitude a nd direction.
Solution :
Magnitude = 2.5×15 = 37.5 m/ s
Since 2.5 is positive, the direction remains south. The new vector is 37.5 m/ s
south.
5.2 W ork ed-Out Example 7 (A dapted)
Question : Find the unit vector in the direction of
?
A = 3
?
i+4
?
j-2
?
k .
Solution :
|
?
A| =
v
3
2
+4
2
+(-2)
2
=
v
9+16+4 =
v
29
4
Page 5


1 Introduction to Ph ysics and Mathematics
Ph ysics is about understanding nature’ s rules, lik e how objects move or how
forces work. Mathematics is the language that mak es these rules clear and pre-
cise. T o describe ph ysical phenomena, we use tools lik e algebr a, trigonome-
try , geometry , vector algebr a, differential calculus, and integr al calculus. This
chapter focuses on vectors, calculus, significant digits, and measurement errors,
which are essential for solving ph ysics problems. A strong gr asp of these mathe-
matical tools helps us describe, predict, and apply ph ysical principles effectively .
K ey Idea : Mathematics provides a clear and precise wa y to express ph ysics con-
cepts, making them easier to understand and apply .
2 V ectors and Scalars
Ph ysical quantities are either scalars or vectors . Scalars, lik e mass or temper a-
ture, are described b y a number and a unit (e.g., 6 k g). They add using regular
arithmetic (e.g., 6 k g + 3 k g = 9 k g). V ectors, lik e velocity or force, need both a mag-
nitude and a direction (e.g., 5 m/s north). V ectors are represented with an arrow
(e.g.,? v ) or bold letters (e.g., v ) and are dr awn as lines with an arrow indicating
direction, where the length represents magnitude.
V ectors add using the triangle rule : dr aw the first vector , place the tail of the
second vector at the head of the first, and the resultant is the vector from the
tail of the first to the head of the second. F or example, if a particle moves 3 m/s
east and 4 m/s north, the resultant velocity is found b y dr awing a r ight triangle,
yielding a magnitude of
v
3
2
+4
2
= 5 m/s at an angle tan
-1
(4/3) ˜ 53
?
north of
east.
1
3 m/ s
Quantities with magnitude and direction but not following the triangle rule (e.g.,
electric current) are not vectors.
K ey Idea : Scalars have only magnitude; vectors have magnitude and direction
and add via the triangle rule.
2.1 Solved Example: V ector A ddition
Question : A cyclist rides at 12 m/ s east, and a wind pushes them at 9 m/ s north.
Find t he resultant velocity’ s magnitude and direction.
Solution : The velocities are perpendicular (? = 90
?
).
|? v| =
v
12
2
+9
2
=
v
144+81 =
v
225 = 15 m/ s
Angle = tan
-1
(
9
12
)
= tan
-1
(0.75)˜ 36.87
?
north of east
The resultant velocity is 15 m/ s at36.87
?
north of east.
2.2 W ork ed-Out Example 1 (A dapted)
Question : A vector has an x-component of 20 units and a y-component of 48
units. Find its magnitude and direction.
Solution : The vector is the resultant of two perpendicular components: 20 units
along x-axis, 48 units along y-axis.
Magnitude =
v
20
2
+48
2
=
v
400+2304 =
v
2704 = 52 units
Angle with x-axis = tan
-1
(
48
20
)
= tan
-1
(2.4)˜ 67.38
?
The vector has magnitude 52 units at 67.38
?
from the x-axis.
3 Equality of V ectors
Two vectors are equal if they have the same magnitude and direction. F or exam-
ple, a velocity of 10 m/s west at one point is equal to 10 m/s west at another point,
2
as long as magnitude and direction are unchanged. Moving a vector par allel to
itself (par allel tr anslation) does not alter it.
K ey Idea : Equal vectors have identical magnitudes and directions, regardless of
position.
3.1 Solved Example: V ector Equality
Question : Are two forces
?
F
1
= 25 N at45
?
north of east and
?
F
2
= 25 N at45
?
north
of e ast equal?
Solution : Both have the same magnitude (25 N) and direction (45
?
north of east).
Thus,
?
F
1
=
?
F
2
.
4 A ddition of V ectors
T o add vectors? a and
?
b , use the triangle rule : dr aw? a , then dr aw
?
b from the head of
? a . The vector from the tail of? a to the head of
?
b is? a+
?
b . Alternatively , the par allelo-
gr am rule dr aws both vectors from a common point, completes a par allelogr am,
and tak es the diagonal through the common tail as the sum.
F or ma gnitudesa ,b , and angle? between them, the resultant’ s m agnitude is:
|? a+
?
b| =
v
a
2
+b
2
+2ab cos?
The anglea with? a is:
tana =
b sin?
a+b cos?
? a
K ey Idea : V ector addition uses geometric rules to combine magnitude and di-
rection.
4.1 Solved Example: V ector A ddition
Question : Two forces
?
F
1
= 30 N at0
?
(x-axis) and
?
F
2
= 40 N at45
?
act on an object.
Find t he resultant force’ s magnitude and direction.
Solution :
|
?
F
1
+
?
F
2
| =
v
30
2
+40
2
+2×30×40 cos45
?
=
v
900+1600+2400×0.707˜
v
4098.8˜ 64.04 N
tana =
40 sin45
?
30+40 cos45
?
=
40×0.707
30+40×0.707
˜
28.28
58.28
˜ 0.485
3
a˜ tan
-1
(0.485)˜ 25.9
?
The re sultant is64.04 N at25.9
?
from the x-axis.
4.2 W o rk ed-Out Example 2 (A dapted)
Question : Find the resultant of three vectors: 6 m at37
?
, 4 m along x-axis, and 3
m al ong y-axis.
Solution : Resolve into x and y components. - 6 m at 37
?
: x = 6 cos37
?
˜ 4.8 m,
y = 6 sin37
?
˜ 3.6 m. - 4 m along x-axis: x = 4 m, y = 0 . - 3 m along y-axis: x = 0 ,
y = 3 m.
T otal x-component: 4.8+4+0 = 8.8 m. T otal y-component: 3.6+0+3 = 6.6 m.
Magnitude =
v
8.8
2
+6.6
2
˜
v
77.44+43.56˜ 11 m
Angle = tan
-1
(
6.6
8.8
)
˜ 36.87
?
The r esultant is approximately 11 m at 36.87
?
.
5 M ultiplication of a V ector b y a Number
Multiplying a vector? a with magnitudea b y a numberk gives
?
b = k? a , with magni-
tude|k|·a . Ifk > 0 , the direction is unchanged; ifk < 0 , it reverses. F or example,
if? a = 4 m/ s east, then-2? a = 8 m/ s west. A vector can be written as? a = a? u , where
? u is a unit vector (magnitude 1) in? a ’ s direction.
K ey Idea : Scalar multiplication scales a vector ’ s magnitude and ma y reverse its
direction.
5.1 Solved Example: Scalar Multiplication
Question : A velocity? v = 15 m/ s south is multiplied b y 2.5. Find the new vector ’ s
magnitude a nd direction.
Solution :
Magnitude = 2.5×15 = 37.5 m/ s
Since 2.5 is positive, the direction remains south. The new vector is 37.5 m/ s
south.
5.2 W ork ed-Out Example 7 (A dapted)
Question : Find the unit vector in the direction of
?
A = 3
?
i+4
?
j-2
?
k .
Solution :
|
?
A| =
v
3
2
+4
2
+(-2)
2
=
v
9+16+4 =
v
29
4
Unit vector =
?
A
|
?
A|
=
3
?
i+4
?
j-2
?
k
v
29
=
3
v
29
?
i+
4
v
29
?
j-
2
v
29
?
k
6 Subtr action of V ectors
T o find ? a-
?
b , add ? a to -
?
b (same magnitude as
?
b , opposite direction) using the
triangle rule. Dr aw? a , then-
?
b from its head; the resultant is from the tail of? a to
the head of-
?
b . The magnitude is:
|? a-
?
b| =
v
a
2
+b
2
+2ab cos(180
?
-?)
K ey I dea : Subtr action is adding the negative vector using the triangle rule.
6.1 Solved Example: V ector Subtr action
Question : Two displacements are
?
A = 12 m at 60
?
and
?
B = 12 m at 60
?
. Find the
magnitude of
?
A-
?
B .
Solution : Since
?
A and
?
B are identical,-
?
B is 12 m at 240
?
. The angle between
?
A
and-
?
B is180
?
.
|
?
A-
?
B| =
v
12
2
+12
2
+2×12×12 cos180
?
=
v
144+144-288 = 0 m
The result is the zero vector .
6.2 W ork ed-Out Example 8 (A dapted)
Question : If|? a+
?
b| =|? a-
?
b| , show? a?
?
b .
Solution :
|? a+
?
b|
2
=? a·? a+2? a·
?
b+
?
b·
?
b = a
2
+b
2
+2? a·
?
b
|? a-
?
b|
2
= a
2
+b
2
-2? a·
?
b
If equal:
a
2
+b
2
+2? a·
?
b = a
2
+b
2
-2? a·
?
b =? 4? a·
?
b = 0 =? ? a·
?
b = 0
Since? a·
?
b = ab cos? = 0 , anda,b?= 0 , then cos? = 0 =? ? = 90
?
. Thus,? a?
?
b .
5