Scalar products

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?

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

Prove Pythagoras' Theorem for right-angled spherical triangles.

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.

Think about the bond angles occurring in a simple tetrahedral molecule and ammonia.

Explore the lattice and vector structure of this crystal.

Explore the meaning of the scalar and vector cross products and see how the two are related.

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.

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.

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 perpendicular.