Applications
Find a vector orthogonal to both $\langle1,2,3 \rangle$ and $\langle 3,2,1 \rangle$.
What is the area of the fundamental parallelegram of these two vectors?
What is the volume of the fundamental parallelpiped of these three vectors?
Solution
Applications
Prove that
$$(a \times b) \cdot (c \times d) = \left| \begin{array}{cc} a \cdot c & b\cdot c \\ a \cdot d & b \cdot d \end{array} \right|.$$
Applications
Assume $a \neq 0$.
If $a\cdot b = a \cdot c$, does $b = c$?
If $a\times b = a \times c$, does $b = c$?
If both occur, does $b = c$?