Addition and Multiplication of Vectors

Vector addition

Vectors do not follow the ordinary laws of algebra because it posses direction. So, vectors are added geometrically. We need to find the resultant of the vector by adding two or more vector. The resultant of the vector is called composition of a vector. There are two laws of vector addition for adding two vectors. They are:

  1. Triangle law of vector addition
  2. Parallelogram law of vector addition
  3. Polygon law of vector addition

Subtraction of vectors also follow the same rules.

Multiplication of vectors

For, a multiplication of two vectors, there are certain laws.Vectors can be multiplied in two different ways. Vectors are multiplied to each other to give either scalar or vector product.

  1. Scalar Product (dot product) of two vector
  2. Vector Product of two vector

1. Scalar product of two vectors: If A and B are any two non zero vectors, then the scalar product of these vectors is given by:

a.b = |a|.|b|Cosθ

where θ  is the angle between the directions of A and B. The product between the two vectors is represented by a small dot between the two vectors. Hence, it is also called the dot product. It is a scalar quantity.

Properties of scalar product:

  • Scalar product follows commutative rules. i.e.,
    a.b = b.a = abCosθ
  • It obeys distributive law. i.e.,
    a.(b+c) = a.b + a.c
  • If two vectors are perpendicular to each other, then product of two vector will be zero. i.e.,
    a.b = abcos90o = 0
  • If two vectors are either parallel or anti parallel to each other, then the product of two vectors will be equal to product of two unit vector. i.e.,
    a.b = abCos0o or 180 o = ab
  • Square of a vector is equal to the square of its magnitude.
    a2 = a.a = a.acos0o = a2

Examples of scalar product:

Work done by a force is  an example of scalar product. Work done is the scalar product of applied force and displacement given by:

W = F.S = FScosθ

Here θ is the angle between F and S.

Power is another example of scalar product. Power is the scalar product of force and velocity given by:

P = F.v = Fvcosθ

Here, θ is the angle between F and v.

2. Vector product of two vectors: If A and B are any two non zero vectors, then the scalar product of these vectors is given by:

a.b = |a|.|b|Sinθ

where θ  is the angle between the directions of A and B.

  • Vector product does not follow commutative rules. i.e.,
    a×b ≠ b×a
  • It obeys distributive law. i.e.,
    a×(b+c) = a×b + a×c
  • If two vectors are perpendicular to each other, then product of two vector will always be greater than zero. i.e.,
    a×b = absin90o > 0
  • Vector product of two parallel vector is zero vector.
    a×b = absin0o or 180 o = 0
  • Square of a vector is equal to zero.
    a2 = a×a = a×asin0o = 0

Examples of vector product:

Torque or moment of foce is an example of vector product.

T = Frsinθ

or, T = r × F

where, θ is the angle between force and perpendicular distance of line of action of force from  a fixed point.

Angular momentum is another example of vector product. It is the rotational analogue of linear momentum. It is defined as the product of P and distance of line of action of P from origin O given as:

J = PrSinθ

or, J = r × P

Where θ is the angle between r and P.

 

 

 

 

 

 

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