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.