Mathematical Physics Vol 1

Chapter 3. Examples

64

Proof

α a = α ( a x i + a y j + a z k )= α a x i + α a y j + α a z k .

Problem14

Prove that

a · b = a x b x + a y b y + a z b z .

Proof

a · b =( a x i + a y j + a z k ) · ( b x i + b y j + b z k )= = a x b x i · i + a x b y i · j + a x b z i · k + a y b x j · i + a y b y j · j + a y b z j · k + + a z b x k · i + a z b y k · j + a z b z k · k = = a x b x + a y b y + a z b z .

Problem15

Prove that

a x a y a z b x b y b z c x c y c z

a · ( b × c )=

.

Proof Given that a · ( b × c )=( a x i + a y j + a z k ) · (( b y c z − b z c y ) i +( b z c x − b x c z ) j +( b x c y − b y c x ) k )= = a x ( b y c z − b z c y )+ a y ( b z c x − b x c z )+ a z ( b x c y − b y c x )= determinant . In the proof we used i × i = 0 , i × j = k , i × k = − j , j × j = 0 , j × k = i , j × i = − k , k × k = 0 , k × i = j , k × j = − i .

Problem16

Prove that

| a × b | = q ( a · a )( b · b ) − ( a · b ) 2 .