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