the angle between (Fig . ). If either then θ is not defined, and in this case, we define Observations . is a real number.
. Let be two nonzero vectors, then if and only if are perpendicular to each other. i.e. .
If θ = , then In particular, as θ in this case is . . If θ = π , then | | | | = − In particular, , as θ in this case is π . .
In view of the Observations and , for mutually perpendicular unit vectors ˆ ˆ k we have ˆ ˆ ˆ ˆ i i j j ⋅= = ˆ ˆ , k k ˆ ˆ j k = ˆ ˆ k i ⋅= . The angle between two nonzero vectors is given by or . The scalar product is commutative. i.e.
(Why?) Two important properties of scalar product Property (Distributivity of scalar product over addition) Let be any three vectors, then Fig . (i) B C A l B l A C (ii) A B C l (iv) l C B A (iii) θ θ θ θ p p p p ( < < ) θ ( < < ) θ ( < < ) θ ( < < ) θ Property Let be any two vectors, and l be any scalar. Then If two vectors are given in component form as a i a j a k b i b j b k , then their scalar product is given as ) ( a i a j a k b i b j b k a i b i b j b k a j b i b j b k a k b i b j b k ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a b i i a b i a b i k j i j j j k