Vector Product

Because of its applications to physics, we shall define still another operation with vectors, which is again called a product. This product is only defined in case the vectors to be multiplied are vectors in 3-space, and its category is that of a vector. For the latter reason, it is called the vector product. It is written \(A \times B\) and is read " \(A\) cross \(B\) ".

Let \(A=\left(a_{1}, a_{2}, a_{3}\right), B=\left(b_{1}, b_{2}, b_{3}\right)\). Then we define \[ A \times B=\left(a_{2} b_{3}-a_{3} b_{2}, a_{3} b_{1}-a_{1} b_{3}, a_{1} b_{2}-a_{2} b_{1}\right) \]

Exercises

  1. \(A \times B=-(B \times A)\).
  2. \(A \times(B+C)=(A \times B)+(A \times C)\).
  3. \((a A) \times B=a(A \times B)=A \times(a B)\)
  4. \((A \times B) \times C=(A\boldsymbol{\cdot} C) B-(B \boldsymbol{\cdot} C) A\)
  5. \(((A \times B) \times C)+((B \times C) \times A)+((C \times A) \times B)=0\)
  6. \(A \times B\) is perpendicular to both \(A\) and \(B\).
  7. \((A \times B)^{2}=A^{2} B^{2}-(A \boldsymbol{\cdot} B)^{2}\). Use this to give a geometric meaning to \(|A \times B|\).