Mathematical Methods for Physics and Engineering - Matematica.NET

7.6 MULTIPLICATION OF VECTORSbOθ b cos θaFigure 7.8 The projection of b onto the direction of a is b cos θ. The scalarproduct of a and b is ab cos θ.in the previous example, the speed of the second particle relative to the first isgiven byu = |u| = √ (−3) 2 +(−8) 2 = √ 73.A vector whose magnitude equals unity is called a unit vector. The unit vectorin the direction a is usually notated â and may be evaluated asâ = a|a| . (7.14)The unit vector is a useful concept because a vector written as λâ then has magnitudeλ and direction â. Thus magnitude and direction are explicitly separated.7.6 Multiplication of vectorsWe have already considered multiplying a vector by a scalar. Now we considerthe concept of multiplying one vector by another vector. It is not immediatelyobvious what the product of two vectors represents and in fact two productsare commonly defined, the scalar product and the vector product. As their namesimply, the scalar product of two vectors is just a number, whereas the vectorproduct is itself a vector. Although neither the scalar nor the vector productis what we might normally think of as a product, their use is widespread andnumerous examples will be described elsewhere in this book.7.6.1 Scalar productThe scalar product (or dot product) of two vectors a and b is denoted by a · band is given bya · b ≡|a||b| cos θ, 0 ≤ θ ≤ π, (7.15)where θ is the angle between the two vectors, placed ‘tail to tail’ or ‘head to head’.Thus, the value of the scalar product a · b equals the magnitude of a multipliedby the projection of b onto a (see figure 7.8).219