7.6 MULTIPLICATION OF VECTORS
abO
θbcosθFigure 7.8 The projection ofbonto the direction ofaisbcosθ. The scalar
product ofaandbisabcosθ.in the previous example, the speed of the second particle relative to the first is
given by
u=|u|=√
(−3)^2 +(−8)^2 =√
73.A vector whose magnitude equals unity is called aunit vector.The unit vectorin the directionais usually notatedaˆand may be evaluated as
aˆ=a
|a|. (7.14)
The unit vector is a useful concept because a vector written asλaˆthen has mag-
nitudeλand directionˆa. Thus magnitude and direction are explicitly separated.
7.6 Multiplication of vectorsWe have already considered multiplying a vector by a scalar. Now we consider
the concept of multiplying one vector by another vector. It is not immediately
obvious what the product of two vectors represents and in fact two products
are commonly defined, thescalar productand thevector product. As their names
imply, the scalar product of two vectors is just a number, whereas the vector
product is itself a vector. Although neither the scalar nor the vector product
is what we might normally think of as a product, their use is widespread and
numerous examples will be described elsewhere in this book.
7.6.1 Scalar productThe scalar product (or dot product) of two vectorsaandbis denoted bya·b
and is given by
a·b≡|a||b|cosθ, 0 ≤θ≤π, (7.15)whereθis the angle between the two vectors, placed ‘tail to tail’ or ‘head to head’.
Thus, the value of the scalar producta·bequals the magnitude ofamultiplied
by the projection ofbontoa(see figure 7.8).