Vectors are such a powerful tool in mathematics and physics, that many results can be proved very easily and intuitively.
Statement : If the diagonals of a parallelogram are congruent, then it is a rectangle.
Let be the parallelogram. Let be the origin. So, the position vectors will be from . Diagonals are and . We have .
Therefore, . By parallelogram law of vector addition, . So,
This implies , or is perpendicular to . Hence, is a right angle. This being a parallelogram, will have all other angles equal to and hence it is a rectangle.
Statement : The diagonals of a kite are at right angles.
Let be the kite, with as the origin. Clearly and are equal sides. In terms of vectors,
Canceling common terms,
Note that and are equal. So, . So,
Hence, . Hence the diagonals are perpendicular.
Statement : If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular then the edges in the third pair are also perpendicular.
In a tetrahedron, each triangle shares an edge with the other. Considering any 2 triangular faces, we are left with only 1 edge. The pair of common edge and the uncommon edge is said to be a pair of opposite edges. Let be a tetrahedron. So, , and are the pairs of opposite edges. Let any 2 of them be perpendicular.
and . Therefore,
Expanding the brackets and then adding the equations,
This gives i.e. . Hence is perpendicular to and these 2 form the third pair.