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CHAPTER 7 : VECTORS AND POLAR COORDINATES

7.1 Scalars and Vectors

A Scalar :- It is a quantity which can be fully expressed by its magnitude (size) e.g. mass, length, time, temperature and any real number.

Operations with scalars follow the same rules as in elementary Algebra.

A Vector :- It is quantity having both magnitudes (size) and direction and it obeys the laws of commutation and association in addition . e.g. displacement, velocity, acceleration, force etc.

Directed Line Segment :- It is a line segment having a specific direction . Its direction is shown by an arrow head.

If the direction is from point A to B, the directed line segment is denoted by .

Here A = initial point   and   B = terminal point of

Representation of a Vector

A geometric vector is a quantity that can be represented by the directed line segment with initial point A and terminal point B ( as shown in the figure1 ) . Its magnitude is the length of the directed line segment proportional to the magnitude of the vector . The magnitude is sometimes called the norm .

Analytically a vector is represented by a symbol or a bold face letter U. The magnitude of AB is written as | | or l (AB) or simply AB. The magnitude of U is written as | U | or simply U.

Zero or Null Vector :- It is a vector of magnitude zero . It is written as 0.

It has no specific direction . Graphically it is a point .

If = 0 then | | = 0 and A and B coincides.

Equality of Vectors :- Two vectors a and b whose magnitudes are equal i.e. a = b and which are in the same direction regardless of their initial points, are called equal vectors.

It can be seen that for three vectors a, b and c,

r = a + b + c = ( a + b) + c = a + (b + c) + (a + c) + b

This result can be extended to any number of vectors.

Multiplication of Vectors by Scalars :- (A) Let m be a scalar and a be a vector then the product ma whose magnitude in |m| times of | a | and which is similarly or oppositely directed to a depending upon whether m is positive or negative.

Then

  1. for m > 0, m a is a vector of magnitude m |a| and is in the direction of a .

  2. for m = 0, ma is a null vector.

  3. for m < 0, ma is the vector of magnitude - m |a| and is exactly opposite to the vector a

From this, it follows that (-1) a = -a and 0 a = 0

(B) Other operations with scalars m and n are

(m + n ) a = m a + n a

m (a + b ) = m a + m b

mn (a) = n (ma) = (m n a)

Linear Combination :- Let a and b be two vectors and x and y be any real numbers. Then the vector r = xa + yb is called the linear combination of vectors a and b for example 3a + 4b , -8a + 7b . It can be extended to any finite number of vectors.

Collinear Vectors :- Two vectors a and b are said to be collinear, if they are parallel to one and the same line. This clearly indicates that 1) they must be coincident or (2) they must be parallel. Also, it must be kept in mind that if two vectors are expressed as the scalar multiple of each other then they must be collinear. i.e. if a = xb then a and b are collinear if x ¹ 0 scalar.

Coplanar Vectors :- If vectors are parallel to a given plane, then they are coplanar. Note that if a and b be two non-collinear vectors which can be uniquely expressed as r = xa + yb, x , y ¹ 0 scalars, then they must be coplanar.

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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