Scalar product and Properties of Scalar Product

Scalar product

Definition 8.16

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.

Geometrical meaning of scalar product (projection of one vector on another vector)

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 .

Proof

Note 8.5

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 .

EXERCISE 8.3