Begin2.DVI

I7-45. ~rc=

∑n

j=1

mj~rj

∑n

j=1

mj

Note that this is a weighted sum of the vectors~ri where the

weight factors arem 1 ,m 2 ,...,mn.

I7-46. We have verified the triple scalar productA~·(B~×C~) =B~·(C~×A~) =C~·(A~×B~)

Change symbols and writeX~·(C~×D~) =D~·(X~×C~)SubstituteX~ =A~×B~ to show

(A~×B~)·(C~×D~) =D~

{

(A~×B~)×C~

}

Next one can employ the triple vector product

relation

(A~×B~)×C~=B~(A~·C~)−A~(B~·C~)

to obtain

(A~×B~)·(C~×D~) = (A~·C~)(B~·D~)−(A~·D~)(B~·C~)

I7-48. (a) At extremum value forE(α,β) =

∑N

i=1

(αxi−β−yi)^2 require that

∂E

∂α

=

∑N

i=1

2(αxi−β−yi)xi= 0

∂E

∂β

=

∑N

i=1

2(αxi−β−yi)(−1) = 0

The above equations can be expressed in the form

α

∑N

i=1

x^2 i−β

∑n

i=1

xi=

∑N

i=1

xiyi

α

∑N

i=1

xi−β

∑N

i=1

1 =

∑N

i=1

yi where

∑N

i=1

1 =N

Solve this system of equations to obtain desired result.

(b) y= 7 + 2x

Solutions Chapter 7