Explore the lattice and vector structure of this crystal.
Think about the bond angles occurring in a simple tetrahedral
molecule and ammonia.
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?
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
An account of multiplication of vectors, both scalar products and
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.
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.
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.