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.

A Knight's Journey

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

V-P Cycles

Age 16 to 18 Challenge Level:

Suppose the vector product ${\bf a} \times {\bf b}\neq {\bf 0}$. Define a sequence of vectors ${\bf b_0},\ {\bf b_1},\ {\bf b_2}\ldots $ by ${\bf b_0}={\bf b}$ and ${\bf b_{n+1}}={\bf a}\times {\bf b_n}$

Show that ${\bf b_n} \rightarrow 0$ as $n\rightarrow \infty$ if the length $|{\bf a}|$ is less than one.

If $|{\bf a}|=1$ and $|{\bf b_1}|=r$ find the directions of the first six vectors in the sequence in relation to the vector ${\bf a}$ and draw a diagram showing these vectors. What happens to the sequence? Describe the surface on which the sequence of vectors from ${\bf b_1}$ onwards lies.

Note: You need to know that the vector product ${\bf a} \times {\bf b}$ is the product of the magnitudes of the vectors times the sine of the angle between the vectors and it is a vector perpendicular to ${\bf a}$ and ${\bf b}$.