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7.2 Algebra of Vectors

(1) Addition of two Vectors

  1. Triangle Law :- The sum or resultant of two vectors a and b is the vector formed by placing the initial point of b on the terminal point of a and then joining the initial point of a to the terminal point of b .

    This sum is written as r = a + b

    This law is also known as Tail - Tip rule

  2. Parallelogram Law :- It states that the sum or resultant of a and b is R is the diagonal of the parallelogram for which a and b are adjacent sides. All vectors a, b and R are concurrent as shown in figure 4.

  3. Vector addition is commutative i.e. a + b = b+ c Extension to sums of two or more vectors is immediate.

  4. Vector addition is associative. Observe the adjoining fig.5 we have using tail-tip rule, for triangle for triangle ABC, for triangle OBC, c i.e. r = ( a + b) + c . Also, for triangle OAC , i.e r = a + (b + c).

    Thus , r = ( a + b ) + c = a + (b + c).

2) Subtraction of a Vector :- It is accomplished by adding the negative of vectors i.e. a - b = a + ( - b). Note also that - ( a + b ) = - a - b .

3) Composition of Vectors :- It is the process of determining the resultant of a system of vectors. For this we have to draw a vector polygon, placing tail of each vector at the tip of the preceding vector. Then draw a vector from the tail of the first vector to tip of the last vector in the system. Later on we will show that not all vector systems reduce to a single vector. Also note that the order of vectors are drawn is immaterial.

Free Vector :- A vector which can be shifted along its line of action or parallel to itself and whose initial point can be anywhere, is called a free vector.

Unless stated, we always refer a vector as a free vector .

Localized Vector :- It is also known as a sliding vector. It can be shifted only along its line of action.

By the principle of transmissibility the external effects of this vector remains the same.

Bounded Vector :- It is called as a fixed vector. It must remain at the same point of action or application.

Negative of a Vector :- The negative of a vector u is the vector -u which has the same direction and inclination but is of the opposite sense (of direction).

* The resultant of a system of vectors is the least number of vectors that will replace the given system.

Position Vector of a Point :- If A and B are any two points then is called the position vector of point B with reference to point A.

Usually we fix a point O and define the position vector of points with reference to O which is called the origin of reference.

We shall use a, b, c etc. to denote position vectors of points A, B, C etc. respectively, relative to origin O.

Unit Vector :- A vector in a given direction having unit magnitude is called a unit vector in that direction .

It is usually denoted by etc ( Read as n Cap). Also, note that if U is given vector then the unit vector in the direction (or along or parallel ) to u is given by

Index

7.1 Scalers & Vectors
7.2 Algebra of Vectors
7.3 Representation of a vector in a plane
7.4 Dotor Scalar product
7.5 Polar Co-ordinates
Supplementary Problems

Chapter 8

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