Vector Addition
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
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