Scalar products

article
Multiplication of vectors

An account of multiplication of vectors, both scalar products and vector products.
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
Air routes

Age

16 to 18

Challenge level

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.
problem
Walls

Age

16 to 18

Challenge level

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.
problem
Pythagoras on a sphere

Age

16 to 18

Challenge level

Prove Pythagoras' Theorem for right-angled spherical triangles.
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
Bond angles

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

Explore the lattice and vector structure of this crystal.
problem
Cubestick

Age

16 to 18

Challenge level

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