Monday Math 14
Suppose we have a parallelogram, with two sides of length a and two sides of length b. We can represent the parallel sides of length a with a vector 
and the other two sides with a vector 
. Now let us consider the diagonals. We see that the diagonals of the parallelogram correspond to 
and 
. Using this, what can we say about the sum of the squares of the lengths of the diagonals?
The square of the length of one diagonal is 
, the other is 
. Now, remember for any inner product space, which standard real geometric vectors are, for any vector 
, we have 
. Thus
\cdot(\mathbf{\vec{a}}+\mathbf{\vec{b}})+(\mathbf{\vec{a}}-\mathbf{\vec{b}})\cdot(\mathbf{\vec{a}}-\mathbf{\vec{b}}))
+(\mathbf{\vec{a}}\cdot\mathbf{\vec{a}}-\mathbf{\vec{a}}\cdot\mathbf{\vec{b}}-\mathbf{\vec{b}}\cdot\mathbf{\vec{a}}+\mathbf{\vec{b}}\cdot\mathbf{\vec{b}}))
which is the sum of the squares of the lengths of the four sides. This geometric identity is called the parallelogram law.
The parallelogram law, in the vector form
is also important in linear algebra: for any normed vector space, the space is an inner product space with the norm derived from the inner product in the usual manner if and only if the parallelogram law, as written above, holds for all vectors and 
in the normed space.
Tags: Math, Monday Math, Linear Algebra, Geometry
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