### Inner Product Spaces: Inner product, length, and angle

### Inner products on real vector spaces

For an indication of the origin of the concept of inner product, we look at the plane.

In the plane, a triangle two of whose sides correspond to the vectors #\vec{a}# and #\vec{b}#, the third side has length #\parallel \vec{a}-\vec{b}\parallel#. If the angle between the vectors #\vec{a}# and #\vec{b}# is equal to #\varphi#, then the *law of cosines* tells us that

\[ \norm{\vec{a}-\vec{b}}^2 =\norm{\vec{a}}^2 +

\norm{\vec{b}}^2 -2\,\norm{\vec{a}}\cdot\norm{\vec{b}}\cdot

\cos(\varphi)

\] The term #\norm{\vec{a}}\cdot\norm{\vec{b}}\cdot\cos(\varphi)# on the right-hand side is the inner product of two vectors #\vec{a}# and #\vec{b}# in the plane. For #\vec{a}=\vec{b}#, this term gives the square of the length of #\vec{a}#. The angle between two vectors is also built into this inner product. In the theory of inner products in abstract vector spaces, we cannot start with such an explicit formula nor do we have a definition of length, so the above expression does not make sense. Therefore, we start from certain properties that this inner product on the plane appears to have: symmetry in the two arguments #\vec{a}# and #\vec{b}#, linearity in each of the arguments #\vec{a}# and #\vec{b}# separately, and positivity of the expression if #\vec{a}=\vec{b}\ne\vec{0}#. This gives us a hold on the definition of inner product for abstract vector spaces.

Inner product Let #V# be a real vector space. An **inner product** on #V# is a function that assigns to every pair of vectors #\vec{a},\vec{b}# from #V# a real number #\dotprod{\vec{a}}{\vec{b}}# in such a way that the following three properties are satisfied.

**bilinearity**: #\dotprod{\vec{a}}{\vec{b}}# is linear in both #\vec{a}# and #\vec{b}#:

\[\begin{array}{rcl}

\dotprod{(\lambda \vec{v}+\mu \vec{w})}{\vec{b}}&= &\lambda \cdot(\dotprod{\vec{v}}{\vec{b}})+\mu\cdot (\dotprod{\vec{w}}{\vec{b}})\\

\dotprod{\vec{a} }{(\lambda \vec{v}+\mu \vec{w})} & =&\lambda\cdot (\dotprod{\vec{a}}{ \vec{v}})+\mu \cdot(\dotprod{\vec{a}}{\vec{w}})

\end{array}

\] for all scalars and vectors;**symmetry**: #\dotprod{\vec{a}}{\vec{b}} = \dotprod{\vec{b}}{\vec{a}}# for all #\vec{a},\vec{b}\in V#;**positive-definiteness**: #\dotprod{\vec{a}}{\vec{a}}\geq 0# for all #\vec{a}\in V#, and #\dotprod{\vec{a}}{\vec{a}} = 0# if and only if #\vec{a}= \vec{0}#.

A real vector space with an inner product is often referred to as a **(real) inner product space**.

Since, for each vector #\vec{a}# in an inner product space, #\dotprod{\vec{a}}{\vec{a}}# is a non-negative real number, the expression #\sqrt{\dotprod{\vec{a}}{\vec{a}}}# is well defined. We use it to define length:

Length and distance

In an inner product space the **length** or **norm** of a vector #\vec{a}# is defined by

\[ \norm{\vec{a}}=\sqrt{\dotprod{\vec{a}}{\vec{a}}} \] The **distance** between the vectors #\vec{a}# and #\vec{b}# is defined as the length of the difference vector #\vec{a}-\vec{b}#, that is to say, #\norm{\vec{a} -\vec{b}}#.

The formula below shows that an inner product is uniquely determined by the length.

Polarisation formula In an inner product space, the following formula holds for all vectors #\vec{a}# and #\vec{b}#.

\[ \dotprod{\vec{a} }{\vec{b}}=\frac12\left(\norm{\vec{a}+\vec{b}}^2-\norm{\vec{a}}^2-\norm{\vec{b}}^2 \right)\]

The value of the inner product can be found as follows:

\[\begin{array}{rcl}

\dotprod{\vec{x}}{\vec{y}}&=&\dotprod{\rv{0,-2,5}}{\rv{-3,-1,-2}}\\

&&\phantom{xx}\color{blue}{\text{vectors written out}}\\

&=&0\cdot (-3)-2\cdot(-1)+5\cdot (-2)\\

&&\phantom{xx}\color{blue}{\text{definition of standard inner product}}\\

&=&-8\\

&&\phantom{xx}\color{blue}{\text{simplified}}

\end{array}\]

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