Vector algebra

problem
Flexi quad tan

Age

16 to 18

Challenge level

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.
problem
Quaternions and rotations

Age

16 to 18

Challenge level

Find out how the quaternion function G(v) = qvq^-1 gives a simple algebraic method for working with rotations in 3-space.
problem
Quaternions and reflections

Age

16 to 18

Challenge level

See how 4 dimensional quaternions involve vectors in 3-space and how the quaternion function F(v) = nvn gives a simple algebraic method of working with reflections in planes in 3-space.
problem
Vector journeys

Age

14 to 18

Challenge level

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?
problem
Flexi quads

Age

16 to 18

Challenge level

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?
problem
Three by one

Age

16 to 18

Challenge level

There are many different methods to solve this geometrical problem - how many can you find?
problem
Tetra perp

Age

16 to 18

Challenge level

Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.