Mathematical Physics Vol 1

3.1 Vector algebra

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Proof Given that the square of the magnitude of the vector product, according to definition (1.19), is equal to | a × b | 2 = | a | 2 | b | 2 sin 2 γ , and the scalar product of vectors a and b , according to definition (1.12), is equal to a · b = | a || b | cos γ i | a || a | = | a | 2 , we finally obtain | a × b | 2 = | a | 2 | b | 2 sin 2 γ = | a | 2 | b | 2 1 − cos 2 γ =( a · a )( b · b ) − ( a · b ) 2 .

Problem17 Prove that all vectors, which are linearly dependent of vectors a 1 and a 2 , lie in the plane OM 1 M 2 , if point O is the start, and M 1 and M 2 are the end of vector a 1 and a 2 , respectively.

Proof Two arbitrary scalars α and β define a vector a that is the linear combination of vectors a 1 and a 2 . a = α a 1 + β a 2 . Take a vector b perpendicular to the plane OM 1 M 2 . Given that a 1 and a 2 lie in that plane it follows that b · a 1 = 0 and b · a 2 = 0. Observe now the following scalar product b · a = b · ( α a 1 + β a 2 )= b · α a 1 + b · β a 2 = α b · a 1 + β b · a 2 = α 0 + β 0 = 0 . Given that it is also equal to zero, it follows that the vector b is normal to the vector a and that consequently vector a lies in the plane OM 1 M 2 .

Problem18 Prove that any four vectors in a 3–D space are linearly dependent.

Problem19 Prove that a · b × c = 0 is the necessary and sufficient condition for the vectors a , b and c to lie in the same plane (to be coplanar).