Vector Addition

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  • Topic: Linear algebra, Vector space, Euclidean space
  • Pages : 20 (3189 words )
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  • Published : February 25, 2013
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Directed Line Segment:

VECTORS

• A line segment with direction is called a directed line
segment.
• If ‘A’ and ‘B’ are two distinct points in the space, then the ordered pair (A, B) is called as a directed line segment and is denoted by """""# .
!
• In """""# , ‘A’ is called initial point and ‘B’ is called terminal !
point of """""# .
!

• The distance from ‘A’ to ‘B’ is called the length or magnitude of """""# . The length or magnitude of """""# !
!
is denoted by '"""""# '. Thus '"""""# ' = ! .
!
!
• The direction of """""# is from ‘A’ to ‘B’ or towards ‘A’ to ‘B’. !
• A line which is having the directed line segment is called the support of the directed line segment. • The support of """""# is denoted by ,""""# .
!
!
• Thus every directed line segment has three attributes, namely, direction, magnitude and support. • Two directed line segments are said to be they are having same direction if their supports are parallel and the terminal points lie in the same half plane determined by the line passing through the initial points.

• """""# and """""# are said to have the same direction if their supports are parallel and !
./

1. A≠C and ‘B’ and ‘D’ are lies in the same half plane determined by the line ,""""# . .
2. A=C and A, B, C and ‘D’ are collinear such that ‘B’ and ‘D’ lie on the same ray originating from ‘A’.
"""/
• """""# and .""# are said to have the opposite direction if their supports are parallel but not have the !
same direction.
• """""# and """""# have the opposite direction.
!
!

• If two directed line segments are not containing parallel supports then they are said to have neither same direction nor opposite direction.
Equivalence Class of a directed line segment:
• Two directed line segments """""# and """""# are said to be equivalent if they have the same direction !
./
and same magnitude.
• """""# and """""# are equivalent then we can write them as """""# ~ ./. !
./
! """""#

• Let ‘S’ be the set of all directed line segments in the space. • The set of all equivalent directed line segments is called equivalence class.

"""""# """/ !
• The equivalence class of """""# = 6./ ⃒ .""# ~"""""# 8 ⊆S. !

Physical Quantities:
• The quantities which can be measured are called physical quantities. • Physical quantities are of two types :(a) Scalars and (b) Vectors. • Scalars are those physical quantities which need only magnitude to express them. They may have direction but not needed for their expression.

• Examples: Mass, energy, work, distance, speed, pressure, Current, Time, Area etc.

• A scalar quantity is represented by a real number along with a suitable unit. • Vectors are those physical quantities which need both magnitude and direction to express them & they should obey vector rules (i.e., parallelogram law of addition). • Examples: Velocity, Displacement, Acceleration, Force, Momentum, Current density, Area, etc. • Generally, vectors are denoted by letters with bar (→) such as D E, F … … …etc. or Thick letters #, "# #,

such as H, I, J, … … …etc.
etc.

Note: Area is both scalar & vector, classified according to application. (In general cases area is considered as a scalar in Electric field, magnetic field, considered as a vector.]. Remember vector does not obey ordinary algebra.

In order to measure many physical quantities, such as force or velocity, we need to determine both a magnitude and a direction. Such quantities are conveniently represented as vectors. vectors
Vectors are used to represent quantities that have both a magnitude and a direction

Good examples of quantities that can be represented by vectors are force and velocity. Both of these have a direction and a magnitude.
Let’s consider force for a second. A force of say 5 Newtons that is applied in a particular direction can be applied at any point in space. In other words, the point where we apply the force does not change the force itself. Forces are independent of the point of...
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