Angle Between Vectors And Magnitude of Vectors

Angle Between Vectors And Magnitude of Vectors
VECTORS - TYPES OF VECTORS
Physical quantities : The quantities that are measurable are called physical quantities
    Ex: Length, Mass, Time, Velocity, Force, etc..
    Physical quantities are mainly classified into three types.  a) Scalars  b) Vectors c) Tensors
Vectors: Vectors are those which have both magnitude and direction and also obey laws of vector addition.  
    Ex : velocity, force, momentum ,torque etc
Scalars: Scalars are those which have only magnitude. 
    Ex : mass, time, distance,flux  etc
    Note :  A physical quantity having magnitude and direction but not obeying laws of vector addition is treated as a scalar.
    Ex : Electric current is a scalar quantity.
Representation:
    A vector is graphically represented by a directed line segment. The length of the line segment is proportional to the magnitude of vector and arrow points along the direction.

Modulus of a vector:
The length or magnitude of a vector is known as the modulus of vector. If \(\overline a \) is a vector, then length of magnitude is denoted by |\(\overline a \).| This is read as Mod \(\overline a \)
Types of Vectors 
1) Like vectors : Two or more vectors  (representing same physical quantity) are called like vectors if they are parallel to each other, however their magnitudes can be different

2) Unlike vectors : Two vectors (representing same physical quantity) are called unlike vectors if they act in opposite direction however their magnitudes can be different

3)Equal vectors : Two or more vectors (representing same physical quantity) are called equal if their magnitudes and directions are same.

Suppose two trains are running on parallel tracks with same velocity and direction. Then their velocity vectors will be equal vectors.
4) Null vector or  Zero vector : It is a vector whose magnitude is zero and direction is unspecified.
Ex:  a) Displacement after one complete revolution.
b) Velocity of vertically projected body at the highest point.
 c)  velocity of simple pendulum at extreme position
5) Negative Vector : A vector having the same magnitude and opposite direction of a given vector is called negative vector of the given vector.
     
6)Parallel or collinear vectors : Two or more vectors are said to be collinear when they act along the same line or along parallel lines however their magnitudes may be different. Two vectors  and k are parallel vectors

7)Coplanar vectors : Two vectors are always coplanar. Three or more vectors are said to be coplanar if they are parallel to the same plane or lie in the same plane.Their magnitudes can be different

8)Unit vector : A vector which has unit modulus is called unit vector. Let \(\vec a\) be a vector such that \(\vec a\ne\)0, then \(\frac{{\vec a}}{{|\vec a|}}\) is a unit vector having the same direction as \(\vec a\). A unit vector is denoted by  and is read "a cap" or "a hat".              
\(\therefore \,\,\hat a = \frac{{\vec a}}{{|\vec a|}}\)  
 In other words if  is a vector with magnitude a and \(\hat i \) is a unit vector in the direction \(\vec a\) then \(\hat i = \frac{{\vec a}}{a} \Rightarrow \vec a = a\hat i\)     .
Note :  are called perpendicular unit vectors along x,y & z axis directions.
9)Position vector : It is a vector that represents the position of a particle with respect to the origin of a co-ordinate system.  
Position vector in 3D plane:
    A point in space can be specified by a position vector \(\overrightarrow R = x\hat i + y\hat j + z\hat k\), x, y and z being the coordinates of the point in cartesian coordianate system

Magnitude of position vector is given by \(\left| {\overrightarrow R } \right| = \sqrt {{x^2} + {y^2} + {z^2}} \)
    The unit vector along  is given by \(\hat R = \frac{{\overrightarrow R }}{{\left| {\overrightarrow R } \right|}} = \frac{{x\hat i + y\hat j + z\hat k}}{{\sqrt {{x^2} + {y^2} + {z^2}} }}\)
10) Co-initial vectors : The vectors having same initial points are called co-initial vectors.
11)Co-terminal vectors : The vectors having same end points are called co-terminal vectors.
12) REAL VECTOR  or  POLAR VECTOR : If the direction of a vector is independent of the coordinate system, then it is called a polar vector. 
Ex: linear velocity, linear momentum, force etc.
13) PSEUDO VECTOR or axial vector  : Vectors associated with rotation about an axis and whose direction is changed when the co-ordinate system is changed from left to right, are called  pseudo vectors.    
 Example : Torque, Angular momentum, Angular velocity etc.
14)Localized and free vectors : Vectors have two broad categories.
a) Localized vector : A vector drawn   parallel to given vector through a specified point in space is called a localized vector.
Its effect changes with the change in its   position.
 b) Free vector : If the origin of vector is not specified then the vector is called free vector. The effect of such a vector remains uneffected if a change of position occur.Statics deals with free vectors