Use of Vectors to Establish Simple Properties of Geometrical Figures

Use of Vectors to Establish Simple Properties of Geometrical Figures

Vectors and Geometrical Figures

  • Vectors can be used to establish important properties of geometrical figures. This includes figures such as triangles, parallelograms, or hexagons in two dimensions, as well as polyhedra in three dimensions.
  • The positional vectors of the corners or vertices of a geometric figure can provide information about its size, shape, and position in space.

Triangles and Midpoints

  • If two points A and B in the plane or in space have position vectors a and b respectively, then the midpoint M of the line segment AB has a position vector 1/2(a + b) or (a + b)/2.
  • This is easily extended to three points in space, allowing us to find the center of a triangle.

Parallelograms

  • A parallelogram can be defined using vectors. If A, B, and C are three points with position vectors a, b, and c respectively, and if c = a + b, then ABCA forms a parallelogram.
  • The diagonals of a parallelogram bisect each other. If D is the midpoint of the diagonals with position vector d, then d = 1/2(a + b).

Polygons

  • More generally, for a polygon of n points in the plane or in space, the position vector of the centroid or geometric center is the average of the position vectors of the vertices.
  • The vertices of regular polygons in the plane or in space can also be obtained by rotating a given vector by a certain angle.

Polyhedra

  • Similarly, for a polyhedron with n vertices in space, the position vector of the centroid is the average of the position vectors of the vertices.
  • We can use this to establish basic properties of polyhedra such as tetrahedrons, cubes, and more complex platonic and Archimedean solids.

Applications

  • Understanding how vectors can be used in this way is a fundamental part of geometrical reasoning.
  • Such methods of vector algebra and vector geometry form the basis of much practical work in fields as diverse as computer graphics, engineering, and physics.