You may also like

Flexi Quads

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Flexi Quad Tan

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

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?