TEACHING NOTES - [XIII] VECTORS (The following are optional. Teach if required) Sine All
1.
2.
ASTC rule
tan cos
sin A sin B sin C = = a b c
Sine rule
3.
Cosine rule
1.
INTRODUCTION
B
c
a
A
C b
a2 = b2 + c2 – 2bc cos A
VECTORS Scalars: Physical quantities which can be completely described by a numerical value with unit are known as scalars. A scalar can be positive, negative, or zero e.g. mass, temperature, density, charge, etc. [Explain here why they are scalars] Vectors: Any physical quantity which have magnitude and direction, and also follows laws of vector algebra are known as vectors e.g. force, velocity, etc.[ Explain here why they are vectors] [Explain with help of displacement] In these lectures we will learn how to deal with quantities having direction as important property. How to symbolize these quantities. How their addition, subtraction, etc. are different from scalar quantities. 2.
REPRESENTATION OF VECTOR A representation of vector will be complete if it gives us direction and magnitude. Symbolic form: v , a , F, s used to separate a vector quantity from scalar quantities (u, i, m)
.
Graphical form: A vector is represented by a directed straight line, having the magnitude and direction of the quantity represented by it. gth Len
Head
Tail By definition magnitude of a vector quantity is scalar and The size or length of a vector is called its magnitude. The magnitude of a vector can be positive or zero, but it cannot be negative.
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3.
TERMINOLOGY OF VECTORS Parallel vector: If two vectors have same direction, they are parallel to each other. They may be located anywhere in the space.
Antiparallel vectors: When two vectors are in opposite direction they are said to be antiparallel vectors. Equality of vectors: When two vectors have equal magnitude and are in same direction and represent the same quantity, they are equal. i.e. ab
Thus when two parallel vectors have same magnitude they are equal. (Their initial point & terminal point may not be same) Negative of a vector: When a vector have equal magnitude and is in opposite direction, it is said to be negative vector of the former. i.e. or a b b a
Thus when two antiparallel vectors have same magnitude they are negative of each other. 4.
LAWS OF ADDITION AND SUBTRACTION OF VECTORS: Triangle rule of addition: Steps for adding two vectors representing same physical quantity by triangle law. Vector Addition To add B to A
(1)
Draw A
(2)
Place the tail of B at the tip of A
(3)
Draw an arrow from the tail of A to the tip of B . This is vector A + B .
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Note : A vector is not tied to a particular location on the page. You can move a vector around as long as you don’t change its length or the direction it points. Vector B is not changed by sliding it to where its tail is at the tip of A Polygon Law of addition: This law is used for adding more than two vectors. This is extension of triangle law of addition. We keep on arranging vectors such that tail of next vector lies on head of former. When we connect the tail of first vector to head of last we get resultant of all the vectors.
Note: P = ( a b) c d = ( c a ) b d
[Associative Law]
Parallelogram law of addition:
(i)
(ii)
(iii)
AC a b
Note : AC a b and AC b a thus a b = b a [Commutative Law]
Eg. Sol.
Note : Angle between 2 vectors is the angle between their positive directions. Suppose angle between these two vectors is , and | a | = a, | b | = b | a b | = a 2 b 2 2ab cos angle with vector a is tan = bsin/(a+bcos) Two vectors of 10 units & 5 units make an angle of 120° with each other. Find the magnitude & angle of resultant with vector of 10 unit magnitude. | a b | = a 2 b 2 2ab cos = 100 25 2 10 5(1 / 2) = 125 50 = 75 = 5 3
1 5 sin 120 5 3 5 3 = = = 3 10 5 cos120 20 5 5 3 = 30° [Here show what is angle between both vectors = 120° and not 60°] tan =
Some Important Results: (1) If = 0° a || b then, | R | | a | | b | & | R | is maximum (2) If = a anti || b then, | R | | a | | b | & | R | is minimum
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(3)
If = /2 R=
(4)
(5)
a b
a 2 b2
& tan = b/a ( is angle with a ) |a| = |b| = a | R | = 2acos/2 & = /2 If | a | = | b | = a & = 120° then | R | = a
Multiplication of a vector by a number: Let say we have a vector a and k is a number. Vector b = k a is defined as a vector of magnitude |ka|. If k is a positive then direction of b is along a & if k is negative then direction of b is opposite to a . Vector subtraction To subtract B from A
Eg. Sol.
(1)
Draw A
(2)
Place the tail of – B at the tip of A .
(3)
Draw an arrow from the tail of A to the tip of – B . This is vector A B .
Two vectors of equal magnitude 2 are at an angle of 60° to each other find magnitude of their sum & difference. | a b | = 2 2 22 2 2 2 cos 60 = 4 4 4 = 2 3 |a b| =
2 2 2 2 2 2 2 cos120 =
4 44 = 2
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Zero(Null) vector When a = b & if want to find a – b = zero(null) vector. It is a vector with zero magnitude & undefined direction.
5.
UNIT VECTOR: A unit vector is a vector of magnitude of 1, with no units. Its only purpose is to point, i.e. to describe a direction in space. ˆ A unit vector in direction of vector A is represented as A A ˆ = &A |A| ˆ or A can be expressed in terms of a unit vector in its direction i.e. A = | A | A Unit Vectors along three coordinates axes:– unit vector along x-axis is ˆi unit vector along y-axis is ˆj unit vector in z-direction is kˆ
6.
1.
RESOLUTION OF VECTOR: a = acos ˆi + acos ˆj
acos is known as component of a along x-axis (ax) acos is known as component of a along y-axis (ay) Results: Unit vector along a ( aˆ ) Since, a = a aˆ & a = a (ˆi cos ˆj cos )
ayj axi
aˆ = ˆi cos ˆj cos 2.
If components of a vector along x & y-axis are known, then that vector can be completely represented as a = a ˆi a ˆj x
y
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3.
|a| =
4.
ay tan = a x
5.
cos =
Eg.
Find a vector of magnitude 50N parallel to 4ˆi 3ˆj
Eg.
Find a vector of magnitude twice of 12ˆi 5ˆj and anti-parallel to 3ˆi 4ˆj
Eg.
Find total x & y component hence express resultant force as vector in aˆi bˆj format.
a 2x a 2y
where, is angle with x-axis
ax , and cos = a
ay a
are called as direction cosines.
Illustration Just after firing, a bullet is found to move at an angle of 37° to horizontal. Its acceleration is 10 m/s2 downwards. Find the component of acceleration in the direction of the velocity. (A*) – 6 m/s2 (B) – 4 m/s2 (C) – 8 m/s2 (D) – 5 m/s2
POSITION VECTOR: Position vector for a point is vector for which tail is origin & head is the given point itself. Position vector of a point defines the position of the point w.r.t. the origin. OP = r = ˆ ˆ r x i yj
DISPLACEMENT VECTOR: Change in position vector of a particle is known as displacement vector. OP = r = x ˆi y ˆj 1
1
1
OQ = r2 = x 2ˆi y 2ˆj PQ = r2 r1 = ( x 2 x1 )iˆ ( y 2 y1 )ˆj Thus we can represent a vector in space starting from (x1, y1) & ending at (x , y ) as ( x x )ˆi ( y y )ˆj 2
2
2
1
2
1
Q(x2,y2) r2
S
P(x1,y1) r1
s Average velocity vector is defined as v av t
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8.
PRODUCT OF VECTORS: Scalar Product (Dot Product) Dot product of two vectors a and b is defined as A B AB cos where, is angle between them when they are drawn with tails coinciding. For any two vectors A and B , AB cos = BA cos . This means that A · B = B ·A . The scalar product obeys the commutative law of multiplication; the order of the two vectors does not matter. Results:–
2.
= 0 a·b = ab = /2 a·b = 0
3.
ˆi·ˆj = iˆ·kˆ = ˆj·kˆ = 0
4.
ˆi·ˆi = ˆj·ˆj = kˆ·kˆ = 1 a = a x ˆi a y ˆj a z kˆ ˆ ˆ ˆ b = b x i b y j bz k thus, a·b = a x b x a y b y a z b z a ·b cos = this is used to find the angle between two vectors. | a || b |
1.
5.
6.
(Used to test orthogonality)
Illustration-1) If a 3ˆi 4ˆj and b 2ˆi ˆj then find angle between them. a ·b Sol. cos = ; | a | = 5; | b | = 5 | a || b | 64 2 1 2 So, cos = = = cos 5 5 5 5 5 5 .
(a)
(b)
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The scalar product is a scalar quantity, not a vector, and it may be positive, negative, or zero. When is between 0° and 90°, cos > 0 and the scalar product is positive (Fig.1.26a). When is between 90° and 180° so that cos < 0, the component of B in the direction of A is negative, and A · B is negative. Finally, when = 90°, A · B = 0. The scalar product of two perpendicular vectors is always zero..
(a)
(b)
(c)
We will use the scalar product to describe work done by a force. When a constant force F is applied to a body that undergoes a displacement s , the work W (a scalar quantity) done by the force is given by W F· s In later chapters we’ll use the scalar product for a variety of purposes, from calculating electric potential. Calculating the scalar product using components We can calculate the scalar product A·B directly if we know the x-, y-, and z-components of A and ˆi · ˆi ˆj · ˆj kˆ · kˆ = (1) (1) cos 0° = 1 B . We find ˆi · ˆj ˆi · kˆ ˆj · kˆ = (1) (1) cos 90° = 0 Now we express A and B in terms of their components, expand the product, and use these products of unit vectors. A·B = (Ax ˆi + Ay ˆj + Az kˆ ) · (Bx ˆi + By ˆj + Bz kˆ ) AxBx + AyBy + AzBz Finding angles with the scalar product
cos =
A xBx A yB y A zBz A B
AxBx AyBy AzBz
= A 2 A 2 A 2 B 2 B 2 B 2 x y z x y z
Vector Product
The vector product of two vectors A and B , also called the cross product, is denoted by A B . As the name suggests, the vector product is itself a vector. We will use this product to describe torque and angular momentum and extensively to describe magnetic fields forces. C = A × B, then C = AB sin
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Note: the vector product of two parallel or antiparallel vectors is always zero. In particular, the vector product of any vector with itself is zero.
Figure : (a) The vector product A B . determined by the right-hand rule. (b) B A = – A B Note: The vector product is not commutative! In fact, for any two vectors A and B , A B B A
(a) (b) Figure: Calculating the magnitude AB sin of the vector product of two vector, A B . Calculating the Vector Product Using Components ˆi ˆi ˆj ˆj kˆ kˆ 0 Using the right-hand rule, we find ˆi ˆj ˆj ˆi kˆ ; ˆj kˆ kˆ ˆj ˆi ;
kˆ ˆi ˆi kˆ ˆj
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Fig.:
(a) We will always use a right handed coordinate system, like this one. (b) We will never use a left handed coordinate system (in which ˆi ˆj kˆ , and so on.)
A B = A x ˆi A yˆj A z kˆ × B x ˆi B y ˆj Bz kˆ
(A y B z A z B y ) ˆi ( A z B x A x B z )ˆj ( A x B y A y B z )kˆ
Thus the components of C = A B are given by Cx = AyBz – AzBy Cy = AzBx – AxBz
Cz = AxBy – AyBx
The vector product can also be expressed in determinant form as ˆi A B Ax Bx
ˆj Ay By
kˆ Az Bz
Vector product vs. scalar product
Be careful not to confuse the expression AB sin for the magnitude of the vector product A B with the similar expression AB cos for the scalar product A · B . To see the contrast between these two expressions, imagine that we vary the angle between A and B while keeping their magnitudes constant. When A and B are parallel, the magnitude of the vector product will be zero and the scalar product will be maximum. When A and B are perpendicular, the magnitude of the vector product will be maximum and the scalar product will be zero.
SOME BASICS OF MATHS 1.
Small angle approximation sin tan cos 1 –
2.
Binomial approximation
(1 x ) n 1 nx if x << 1 53°
3.
sin 37°, cos 37°
2 1 2
3
5 37°
.
4
{Home Work :
Chaper-2
Ex. Obj.
Q.1 to 18 I (1 to 5) & II (All)
}
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