Properties of Vector Addition
( i ) a + b = b + a ( commutativity ) ( ii ) a + ( b + c )= ( a + b )+ c ( associativity ) ( iii ) a + O = a ( additive identity ) ( iv ) a + (— a ) = 0 ( additive inverse ) ( v ) ( k1 + k2 ) a = k1 a + k2a ( multiplication by scalars ) ( vi ) k ( a + b ) = k a + k b ( multiplication by scalars ) ( vii ) | a + b | ≤ | a | + | b | and | a – b | ≥ | a | – | b |
Difference ( Subtraction ) of Vectors
If a and b be any two vectors , then their difference a – b is defined as a + ( - b ).
Multiplication of a Vector by a Scalar
Let a be a given vector and λ be a scalar . Then , the product of the vector a by the scalar λ is λ a and is called the multiplication of vector by the scalar .
Important Properties
( i ) | λ a | = | λ | | a | ( ii ) λ O = O ( iii ) m ( -a ) = – ma = – ( m a ) ( iv ) ( -m ) ( -a ) = m a ( v ) m ( n a ) = mn a = n ( m a ) ( vi ) ( m + n ) a = m a + n a ( vii ) m ( a + b ) = m a + m b
Vector Equation of Joining by Two Points
Let P1 ( x1 , y1 , z1 ) and P2 ( x2 , y2 , z2 ) are any two points , then the vector joining P1 and P2 is the vector P1 P2 .