Suppose we have two vectors from an orthonormal system, u → {\displaystyle {\vec {\mathbf {u}}}} and v → {\displaystyle {\vec {\mathbf {v}}}} . Taking the inner product of these vectors, we get
u → ∙ v → = ∑ k = 1 3 u k a → k ∙ ∑ m = 1 3 v m a → m = ∑ k = 1 3 u k ∑ m = 1 3 v m a → k ∙ a → m = ∑ k = 1 3 u k ∑ m = 1 3 v m δ k , m = ∑ k = 1 3 v k u k {\displaystyle {\vec {\mathbf {u}}}\bullet {\vec {\mathbf {v}}}=\sum _{k=1}^{3}u_{k}{\vec {\mathbf {a}}}_{k}\bullet \sum _{m=1}^{3}v_{m}{\vec {\mathbf {a}}}_{m}=\sum _{k=1}^{3}u_{k}\sum _{m=1}^{3}v_{m}{\vec {\mathbf {a}}}_{k}\bullet {\vec {\mathbf {a}}}_{m}=\sum _{k=1}^{3}u_{k}\sum _{m=1}^{3}v_{m}\delta _{k,m}=\sum _{k=1}^{3}v_{k}u_{k}}
What if they aren't from a normalized system, so that
and 
. Taking the inner product of these vectors, we get
