Mathematical Physics Vol 1

Chapter 3. Examples

68

Problem22 Determine vector r if its scalar products ( a i = r · A i ) with three noncoplanar vectors A i ( i = 1 , 2 , 3) are given.

Solution The following equations

A 2 × A 3 V

A 3 × A 1 V

A 1 × A 2 V

A ′ 1 =

, A ′ 2 =

, A ′ 3 =

(3.4)

.

determine the reciprocal system of vectors A ′ i ( i = 1 , 2 , 3), where V = A 1 · A 2 × A 3 . Let us decompose vector r in the directions of three noncoplanar vectors A ′ i using the Gibbs 1 formula r =( r · A 1 ) A ′ 1 +( r · A 2 ) A ′ 2 +( r · A 3 ) A ′ 3 = 3 ∑ i = 1 ( r · A i ) A ′ i . Given that, according to the initial assumption a i = r · A i , the previous relation becomes

3 ∑ i = 1

a i A ′ i .

r =

Using the relation for reciprocal vectors (3.4) we finally obtain

a 1 ( A 2 × A 3 )+ a 2 ( A 3 × A 1 )+ a 3 ( A 1 × A 2 ) V .

r =

Problem23 Prove that, if a , b and c are noncoplanar vectors and α a + β b + γ c = 0, then α = β = γ = 0 .

Proof Assume that α̸ = 0. It follows from α a + β b + γ c = 0 that α a = − β b − γ c , that is, a = − β α b − γ α c . It further follows that vector a lies in the plane formed by b i c , which is in contradiction with the initial assumption that the vectors are noncoplanar. Thus, α = 0. It can be proved, by analogy, that β = 0 and γ = 0.

1 Josiah Willard Gibbs (1836-1903). American mathematician, one of the founders of vector analysis, mathematical thermodynamics and statistical mechanics (Elementary Principles in Statistical Mechanics, 1902). His work was of great importance for the development of vector analysis and mathematical physics.