Scalar Product of Two Vectors
If a and b are two non-zero vectors , then the scalar or dot product of a and b is denoted by a * b and is defined as a * b = | a | | b | cos θ , where θ is the angle between the two vectors and 0 < θ < π
( i ) The angle between two vectors a and b is defined as the smaller angle θ between them , when they are drawn with the same initial point .
Usually , we take 0 < θ < π . Angle between two like vectors is O and angle between two unlike vectors is π .
( ii ) If either a or b is the null vector , then scalar product of the vector is zero . ( iii ) If a and b are two unit vectors , then a * b = cos θ . ( iv ) The scalar product is commutative i . e ., a * b = b * a ( v ) If i , j and k are mutually perpendicular unit
vectors i , j and k , then i * i = j * j = k * k = 1 and i * j = j * k = k * i = 0
( vi ) The scalar product of vectors is distributive over vector addition . ( a ) a * ( b + c ) = a * b + a * c ( left distributive )
( b ) ( b + c ) * a = b * a + c * a ( right distributive )
Note Length of a vector as a scalar product
If a be any vector , then the scalar product a * a = | a | | a | cosθ ⇒ | a | 2 = a 2 ⇒ a = | a | Condition of perpendicularity a * b = 0 <=> a ⊥ b , a and b being non-zero vectors .
Important Points to be Remembered
( i ) ( a + b ) * ( a – b ) = | a | 2 2 – | b | 2 ( ii ) | a + b | 2 = | a | 2 2 + | b | 2 + 2 ( a * b ) ( iii ) | a – b | 2 = | a | 2 2 + | b | 2 – 2 ( a * b ) ( iv ) | a + b | 2 + | a – b | 2 = (| a | 2 2 + | b | 2 ) and | a + b | 2 – | a –
b | 2 = 4 ( a * b ) or a * b = 1 / 4 [ | a + b | 2 – | a – b | 2 ]
( v ) If | a + b | = | a | + | b |, then a is parallel to b . ( vi ) If | a + b | = | a | – | b |, then a is parallel to b .