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Vectors-1 Compiled by: Udit Gupta Alumnus MNNIT Allahabad Ex-Manager at Reliance Industries Limited Ex-VidyaMandir Faculty of Physics Cofounder and CFO Elements Azamgarh
Content Physical Quantities Geometrical Interpretation of vectors Types of vectors Multiplication of a vector by a scalar Unit vector
Introduction Physical Quantities Scalars Tensors Vectors Magnitude and a sense of direction but do not follow vector addition. Magnitude but no sense of direction Eg. Mass, Temperature, Distance etc. It can be positive, negative or zero. Eg: - 40 c) It is represented by an English alphabet Magnitude and a sense of direction Eg. Force, Velocity, Momentum etC It is represented as: a or a or simply bold a Magnitude of a a| is a positive quantity Eg. Current, Stress Treat current as a tensor only if such option is given in the paper else treat it as scalar
In tensors, normal rules of algebra are applicable. 5A 2A 3A 2A If a physical quantity is vector, it will be independent of choice of coordinate axes 5A 3A
Geometrical Representation Vectors are geometrically represented by directed line segments. Vectors are free vectors i.e. they can be moved in space without changing its orientation Length of the arrow = magnitude of the vector Direction of arrow = direction of vector. Scale: 1cm=1N 5 cm
Geometrical Representation Vector a' has magnitude 4 units and direction is Y axis. b5 units, direction is-X axis. FI = 6 units, direction is-Y axis.
Types of Vectors Equal vectors: Two vectors having same magnitude and same direction. Parallel vectors: Two vectors having same direction. Anti-parallel vectors: Two vectors in opposite directions. Opposite vectors: Two vectors having same magnitude and opposite directions.
Example Try to figure out what type of vectors are given: Opposite Equal Anti-parallel Parallel
Multiplication of a vector by a scalar Let a vector a be multiplied by a scalar k. If k > 0, vector k is in same direction as of and magnitude k times that of a. If k < 0, vector k is in opposite direction as of and magnitude l kl times that of a 2a -3a
Unit Vector Vector with magnitude unity or one. =IA.IA: is the unit vector in direction of Unit vector a is generally denoted as (a cap) Each direction has a unique unit vector. Unit vector in +X-axis is 1, in +Y-axis is j, in +Z- axis IsK A vector in +X direction can be written as its magnitude multiplied by and -Y respectively. Unit vector does not have any dimension and is dimensionless. i, for -X direction multiplied by - . Similarly this could be done for +Y