Let and be any two non-zero vectors and θ be the included angle of the vectors as in Fig. 8.34.
Their scalar product or dot product is denoted by and is defined as a scalar | . | | | cosθ .
Thus ⋅ = | | | | cosθ .
Since the resultant of ⋅ is a scalar, it is called scalar product. Further we use the symbol dot (‘.’) and hence another name dot product.
Let = , = and θ be the angle between and .
Draw BL perpendicular to OA. From the right triangle OLB
cosθ = OL/OB
Properties of Scalar Product
(i) Scalar product of two vectors is commutative.
With usual definition,
⋅ = | | | |cosθ = | | | |cosθ = ⋅
That is, for any two vectors and b ,
⋅ = ⋅ .
(ii) Nature of scalar product
We know that 0 ≤ θ ≤ π .
Suppose three sides are given in vector form, prove
(i) either sum of the vectors is or sum of any two vectors is equal to the third vector, to form a triangle.
(ii) dot product between any two vectors is 0 to ensure one angle is p/2 .