Mathematical Physics Vol 1

3.1 Vector algebra

61

Problem7

Prove

( a × b ) × ( c × d )= c ( a · b × d ) − d ( a · b × c )= b ( a · c × d ) − a ( b · c × d )

Solution From x × ( c × d )= c ( x · d ) − d ( x · c ) where x = a × b , it follows that ( a × b ) × ( c × d )= c ( a × b · d ) − d ( a × b · c )= c ( a · b × d ) − d ( a · b × c ) . And from ( a × b ) × y = b ( a · y ) − a ( b · y ) where y = c × d , it follows that ( a × b ) × ( c × d )= b ( a · c × d ) − a ( b · c × d ) .

Problem8 Prove that the following equality is true

[ a , b , c ] d +[ a , c , d ] b − [ a , b , d ] c − [ b , c , d ] a = 0 .

Solution Observe first the vector product of two vector products ( a × b ) × ( c × d ) .

This product yields a vector normal to both vectors a × b and c × d . It follows that this vector lies in the plane determined by vectors a , b , but also in the plane determined by vectors c , d , and it can thus be represented in two ways

( a × b ) × ( c × d )=[ a , c , d ] b − [ b , c , d ] a , ( a × b ) × ( c × d )=[ a , b , d ] c − [ a , b , c ] d .

By subtracting these equations we obtain

[ a , b , c ] d +[ a , c , d ] b − [ a , b , d ] c − [ b , c , d ] a = 0 .

Problem9 Prove that three orthogonal vectors in a 3–D space are linearly independent.