Mathematical Physics Vol 1

Chapter 1. Vector algebra

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1.4.4 Scalar (dot or internal) product of two vectors

Definition The scalar product of two vectors a and b , symbolically denoted by a · b (which is read as "a dot b") or ( ab ), is a real number determined by: | a |·| b |· cos( a , b ), i.e. a · b = | a |·| b |· cos γ , (1.12) where γ is the angle between vectors a and b . It follows from the very definition that the scalar product is equal to the projection of the vector a on the direction of the vector b , multiplied by the magnitude (length) of the vector b , i.e. a · b = | b |· proj b a . By analogy, a · b = | a |· proj a b , given the commutativity of the scalar product and the parity of the cos γ function. In mechanics (physics) the scalar product has the following physical meaning. If the force acting on some point M is denoted by S , and the elementary displacement of that point by d r , then the variable d A , defined by the relation d A = S · d r represents the elementary work of the force S on the displacement d r .

γ

b

γ

b

b

γ

a

a

a

(a) a · b > 0 (c) a · b < 0 Figure 1.11: The sign of the scalar product - angle between the vectors (a) sharp (b) right (c) obtuse. (b) a · b = 0

The sign of the scalar product depends on the angle between the vectors. Thus, the product is positive if the angle between vectors is sharp or zero, or if the vectors are orthogonal (right angle), and negative if the angle is obtuse (between π / 2 < γ < π ) (Fig. 1.11). Starting from this definition the magnitude of a vector and the condition under which two vectors are orthogonal can be determined. Namely, in the special case, when a = b , it follows that γ = 0 and, according to (1.12), a · a = | a |·| a |· cos ( a , a )= | a |·| a | = | a | 2 ⇒ | a | = √ a · a . (1.13) Thus, it follows directly from the definition of the scalar product that the square of the vector magnitude is equal to the scalar product of the vector with itself. It also follows from the definition of a scalar product for the angle γ between two vectors

a · b | a |·| b | ⇒

a · b | a |·| b |

cos γ =

γ = arccos

(1.14)

,

and thus for | a |̸ = 0 and | b |̸ = 0 two vectors are orthogonal iff 9 a · b =0. From the previous definitions and properties of real numbers, the following properties, which are also called metric properties of a linear vector space , follow: 9 iff is short for "if and only if" (necessary and sufficient condition).