Scalar products

  • Flexi Quads
    problem
    Favourite

    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?

  • Cubestick
    problem
    Favourite

    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.

  • Air Routes
    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.
  • Pythagoras on a Sphere
    problem

    Pythagoras on a Sphere

    Age
    16 to 18
    Challenge level
    Prove Pythagoras' Theorem for right-angled spherical triangles.
  • Quaternions and Reflections
    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.
  • Bond angles
    problem

    Bond Angles

    Age
    16 to 18
    Challenge level
    Think about the bond angles occurring in a simple tetrahedral molecule and ammonia.
  • Flexi Quad Tan
    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.

  • Coordinated crystals
    problem

    Coordinated Crystals

    Age
    16 to 18
    Challenge level

    Explore the lattice and vector structure of this crystal.

  • Cross with the Scalar product
    problem

    Cross With the Scalar Product

    Age
    16 to 18
    Challenge level

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

  • Walls
    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.

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