Vector Analysis

Vector


. Scalars are the quantities which have magnitude but no direction. For examples mass length, temperatures energy, work etc.

. Vectors are the quantity which have both magnitude and direction and obeys the triangle law of addition or parallelogram law of addition. For example velocity, acceleration, torque etc.,

. Null or Zero Vector is the vector whose origin and terminal point are same or coincide. Its magnitude is zero and direction is indeterminate


. Vector addition follows

Commutative law :

Associative law : =

Distributive law :

If

then



. Subtraction of vectors





is reversed and added to

C =





Vector subtraction

1. Does not obey commutative law


2. Does not obey Associative law


3. Obeys distributive law

Note : Negative vector of a Vector is the Vector of same magnitude but opposite in direction. When a Vector is added to its negative Vector, gives Null Vector.


. Resolution of vectors






Scalar or dot product :

If and are two Vectors and is angle between them,
It is a Scalar



Cross or Vector Product

and are two vectors with an angle between them.




. Vector Relations :

and A + B = C, collinear Vectors, = 0

and A - B = C, collinear Vectors, = 0

are Normal vectors, if = 90^o


. Relative Velocity

Suppose two bodies A and B are moving with velocities

Velocity of A relative to B



tan q =



Change in Velocity :

Initial velocity of a particle is and which changes to in time ''t''. Then change in it's velocity, = where ''q'' is the angle between and . Average acceleration,



A particle moves along a circle with constant speed , wen it has covered angular displacement '''', in time ''t'', then the magnitude of change in velocity is