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Cross with the Scalar Product

Age 16 to 18 Challenge Level:

Consider the vector
{\bf v}=\pmatrix{1\cr 2\cr 3}
Investigate the properties of vectors ${\bf u}$ such that ${\bf u}\cdot {\bf v}=0$.
Describe geometrically the set of all such vectors ${\bf u}$.

Now explore the possibilities for vectors ${\bf w}$ which are the result of taking the vector cross product of ${\bf v}$ with another vector. How does this relate to the first part of the question?

Which of these vectors could arise from taking the vector cross product of ${\bf v}$ with another vector? Before performing lots of algebra, can you work out a quick way to make your decision?
{\bf w}=\pmatrix{0\cr 3\cr -2}\,, \pmatrix{795\cr 11\cr 167}, \pmatrix{1\cr -1\cr 0} \mbox{ or } \pmatrix{-7\cr -7\cr 7}

Can you find a way of quickly constructing other such vectors?