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 $A = (a_{1}, a_{2}, a_{3}), B = (b_{1}, b_{2}, b_{3})$ . Then we define $A \times B = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})$

Exercises

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

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