phy1020.DVI

Appendix O

Vector Arithmetic

A vectorAmay be written in cartesian (rectangular) form as

ADAxiCAyjCA ́k; (O.1)

whereiis aunit vector(a vector of magnitude 1) in thexdirection,jis a unit vector in theydirection, and
kis a unit vector in the ́direction.Ax,Ay, andA ́are called thex,y, and ́components(respectively) of
vectorA, and are the projections of the vector onto those axes.
Themagnitude(“length”) of vectorAis

jAjDAD

q

A^2 xCA^2 yCA^2 ́: (O.2)

For example, ifAD 3 iC 5 jC 2 k, thenjAjDAD

p

32 C 52 C 22 D

p

38.

In two dimensions, a vector has nokcomponent:ADAxiCAyj.

Addition and Subtraction

To add two vectors, you add their components. Writing a second vector asBDBxiCByjCB ́k,wehave

ACBD.AxCBx/iC.AyCBy/jC.A ́CB ́/k: (O.3)

For example, ifAD 3 iC 5 jC 2 kandBD 2 ijC 4 k, thenACBD 5 iC 4 jC 6 k.
Subtraction of vectors is defined similarly:

ABD.AxBx/iC.AyBy/jC.A ́B ́/k: (O.4)

For example, ifAD 3 iC 5 jC 2 kandBD 2 ijC 4 k, thenABDiC 6 j 2 k.

Scalar Multiplication

To multiply a vector by a scalar, just multiply each component by the scalar. Thus ifcis a scalar, then

cADcAxiCcAyjCcA ́k: (O.5)

For example, ifAD 3 iC 5 jC 2 k, then 7 AD 21 iC 35 jC 14 k.