Dot Product
It is possible to multiply a vector by another vector, but there is more than one kind of multiplication between
vectors. One type of vector multiplication is called thedot product, in which a vector is multiplied by another
vector to give ascalarresult. The dot product (written with a dot operator, as inAB)is
ABDABcosDAxBxCAyByCA ́B ́; (O.6)whereis the angle between vectorsAandB. For example, ifAD 3 iC 5 jC 2 kandBD 2 ijC 4 k, then
ABD 6 5 C 8 D 9.
The dot product can be used to find the angle between two vectors. To do this, we solve Eq. (O.6) for
and find cosDAB=.AB/. Applying this to the previous example, we getAD
p
38 andBDp
21 ,so
cosD9=.
p
38p
21/, and thusD71:4ı.
An immediate consequence of Eq. (O.6) is that two vectors are perpendicular if and only if their dot
product is zero.
Cross Product
Another kind of multiplication between vectors, called thecross product, involves multiplying one vector by
another and giving anothervectoras a result. The cross product is written with a cross operator, as inAB.
It is defined by
ABD.ABsin/u (O.7)D
ˇ ˇ ˇ ˇ ˇ ˇ
ijk
Ax Ay A ́
Bx By B ́ˇ ˇ ˇ ˇ ˇ ˇ
(O.8)
D.AyB ́A ́By/i.AxB ́A ́Bx/jC.AxByAyBx/k; (O.9)where againis the angle between the vectors, anduis a unit vector pointing in a direction perpendicular
to the plane containingAandB, in a right-hand sense: if you curl the fingers of your right hand from
AintoB, then the thumb of your right hand points in the direction ofAB(Fig. O.1). As an example, if
AD 3 iC 5 jC 2 kandBD 2 ijC 4 k, thenABD.20.2//i.124/jC. 3 10/kD 22 i 8 j 13 k.
Rectangular and Polar Forms
A two-dimensional vector may be written in eitherrectangular formADAxiCAyjdescribed earlier, or in
polar formADA†, whereAis the vector magnitude, andis the direction measured counterclockwise
from theCxaxis. To convert from polar form to rectangular form, one finds
AxDAcos (O.10)
AyDAsin (O.11)Inverting these equations gives the expressions for converting from rectangular form to polar form:
AD
q
A^2 xCA^2 y (O.12)tanDAy
Ax