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.
Explore the meaning of the scalar and vector cross products and see how the two are related.
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?
An account of multiplication of vectors, both scalar products and
Explore the lattice and vector structure of this crystal.
Think about the bond angles occurring in a simple tetrahedral
molecule and ammonia.
Play countdown with vectors.
Stick some cubes together to make a cuboid. Find two of the angles
by as many different methods as you can devise.
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.
Prove Pythagoras' Theorem for right-angled spherical triangles.
Plane 1 contains points A, B and C and plane 2 contains points A
and B. Find all the points on plane 2 such that the two planes are
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.