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.