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

OrOC = 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 AdditionConsider two vectors A and B as shown in figure. From figure

1. Commutative Law of Vector Addition (A+B = B+A)

OA + AC = OC

Or

A + B = R .................... (1)

And

OB + BC = OC

OrB + 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 VeConsider 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

OrA + (B + C) = R

Hence

(A + B) + C = A + (B + C)

Therefore, vector addition is associative.

Product of Two VectorsScalar Product (Dot Product)

1.

2.Vector Product (Cross Product)

1. Scalar Product OR Dot ProductIf 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 ?}

Examplesof 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 ProductA . (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 ProductWhen 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.

Examplesof 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 ProductArea of Parallelogram = |A x B|

Area of Triangle = 1/2 |A x B|

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