Physics Study Notes..1st Year..Chapter 1
Vector ::::::
Addition of Vectors by Law of Parallelogram
According to the law of parallelogram of addition of vectors, if we
are given two vectors. A1 and A2 starting at a common point O,
represented by OA and OB respectively in figure, then their resultant
is represented by OC, where OC is the diagonal of the parallelogram
having OA and OB as its adjacent sides.
If R is the resultant of A1 and A2, then
R = A1 + A2
Or
OC = OA + OB
But OB = AC
Therefore,
OC = OA + AC
ß is the angle opposites to the resultant.
Magnitude of the resultant can be determined by using the law of cosines.
R = |R| = vA1(2) + A2(2) - 2 A1 A2 cos ß
Direction of R can be determined by using the Law of sines.
A1 / sin ? = A2 / sin a = R / sin ß
This completely determines the resultant vector R.
Properties of Vector Addition
1. Commutative Law of Vector Addition (A+B = B+A)
Consider two vectors A and B as shown in figure. From figure
OA + AC = OC
Or
A + B = R .................... (1)
And
OB + BC = OC
Or
B + A = R ..................... (2)
Since A + B and B + A, both equal to R, therefore
A + B = B + A
Therefore, vector addition is commutative.
2. Associative Law octor Addition (A + B) + C = A + (B + C) f Ve
Consider three vectors A, B and C as shown in figure. From figure
using head - to - tail rule.
OQ + QS = OS
Or
(A + B) + C = R
And
OP + PS = OS
Or
A + (B + C) = R
Hence
(A + B) + C = A + (B + C)
Therefore, vector addition is associative.
Product of Two Vectors
1. Scalar Product (Dot Product)
2. Vector Product (Cross Product)
1. Scalar Product OR Dot Product
If the product of two vectors is a scalar quantity, then the product
itself is known as Scalar Product or Dot Product.
The dot product of two vectors A and B having angle ? between them
may be defined as the product of magnitudes of A and B and the
cosine of the angle ?.
A . B = |A| |B| cos ?
A . B = A B cos ?
Because a dot (.) is used between the vectors to write their scalar
product, therefore, it is also called dot product.
The scalar product of vector A and vector B is equal to the
magnitude, A, of vector A times the projection of vector B onto the
direction of A.
If B(A) is the projection of vector B onto the direction of A, then
according to the definition of dot product.
A . B = A B(A)
A . B = A B cos ? {since B(A) = B cos ?}
Examples of dot product are
W = F . d
P = F . V
Commutative Law for Dot Product (A.B = B.A)
If the order of two vectors are changed then it will not affect the
dot product. This law is known as commutative law for dot product.
A . B = B . A
if A and B are two vectors having an angle ? between then, then their
dot product A.B is the product of magnitude of A, A, and the
projection of vector B onto the direction of vector i.e., B(A).
And B.A is the product of magnitude of B, B, and the projection of
vector A onto the direction vector B i.e. A(B).
To obtain the projection of a vector on the other, a perpendicular is
dropped from the first vector on the second such that a right angled
triangle is obtained
In ? PQR,
cos ? = A(B) / A => A(B) = A cos ?
In ? ABC,
cos ? = B(A) / B => B(A) = B cos ?
Therefore,
A . B = A B(A) = A B cos ?
B . A = B A (B) = B A cos ?
A B cos ? = B A cos ?
A . B = B . A
Thus scalar product is commutative.
Distributive Law for Dot Product
A . (B + C) = A . B + A . C
Consider three vectors A, B and C.
B(A) = Projection of B on A
C(A) = Projection of C on A
(B + C)A = Projection of (B + C) on A
Therefore
A . (B + C) = A [(B + C}A] {since A . B = A B(A)}
= A [B(A) + C(A)] {since (B + C)A = B(A) + C(A)}
= A B(A) + A C(A)
= A . B + A . C
Therefore,
B(A) = B cos ? => A B(A) = A B cos ?1 = A . B
And C(A) = C cos ? => A C(A) = A C cos ?2 = A . C
Thus dot product obeys distributive law.
2. Vector Product OR Cross Product
When the product of two vectors is another vector perpendicular to
the plane formed by the multiplying vectors, the product is then
called vector or cross product.
The cross product of two vector A and B having angle ? between them
may be defined as "the product of magnitude of A and B and the sine
of the angle ?, such that the product vector has a direction
perpendicular to the plane containing A and B and points in the
direction in which right handed screw advances when it is rotated
from A to B through smaller angle between the positive direction of A
and B".
A x B = |A| |B| sin ? u
Where u is the unit vector perpendicular to the plane containing A and
B and points in the direction in which right handed screw advances
when it is rotated from A to B through smaller angle between the
positive direction of A and B.
Examples of vector products are
(a) The moment M of a force about a point O is defined as
M = R x F
Where R is a vector joining the point O to the initial point of F.
(b) Force experienced F by an electric charge q which is moving with
velocity V in a magnetic field B
F = q (V x B)
Physical Interpretation of Vector OR Cross Product
Area of Parallelogram = |A x B|
Area of Triangle = 1/2 |A x B|
No comments:
Post a Comment