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Vectors

HOW TO LOCATE POINTS IN SPACE?

Positions in space are designated relative to Coordinate systems. The Cartesian coordinate system is a particularly convenient coordinate system in which positions are designated by distances (x, y, z) along three perpendicular axes that intersect at a point called the origin.

 diagram  diagram
            In a polar coordinate system positions in a plane are designated by a length r from the origin, and an angle θ usually measured from the positive x – axis. From simple trigonometry we see that the relationships between the polar coordinates and the Cartesian coordinates are

x = r cos θ

y = r sin θ

 diagram

 

A frame of reference is another name for the particular coordinate system with respect to which we are making observations of physical phenomena.

A scalar quantity requires only a number for its complete description. Mass, volume, density, pressure and temperature are all examples of scalar quantities. The mathematics of scalar quantities is the ordinary algebra of numbers.

            Vector quantities require  both magnitude and direction for its complete description. Velocity, acceleration, force and momentum are examples of vector quantities. A vector can be represented graphically by a directed line segment. The length of the line segment represents the vector’s magnitude and its angle with respect to some coordinate system specifies its direction.

 
We will represent vectors by  bold face type letters, with an arrow over the letter such as a. When written by hand the same representation may be used. 
            The magnitude of a vector will be represented by italic type letters such as a, b, c etc.
 diagram
            If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are equal, no matter where they are located in space.  diagram
            The negative of a vector is defined as a vector having the same magnitude as the original vector but the opposite direction.  diagram
            When two vectors a and b  have opposite directions, whether their magnitudes are the same or not, we say that they are antiparallel.  diagram

VECTOR ALGEBRA

 
Addition of Vectors
      (i)   Geometrical Method

Two vectors andb  may be added geometrically by drawing them to a common scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the sum vector c.

 diagram

Vector addition is commutative and obeys the associative law.

 diagram  diagram
      (ii)  Analytical Method

If the two vectors and b are given such that the angle between them is θ. The magnitude of the resultant vector  of their vector addition is given by

equation

and its direction is given by angle Φ with
vector a

equation

 diagram

solution

The post Vectors appeared first on Quantemporary.



This post first appeared on Articles Of Physics Topics For IIT-JEE, PMT, IB, SAT, AP Students, please read the originial post: here

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