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# 14: Orthonormal Bases and Complements

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

You may have noticed that we have only rarely used the dot product. That is because many of the results we have obtained do not require a preferred notion of lengths of vectors. Once a dot or inner product is available, lengths of and angles between vectors can be measured--very powerful machinery and results are available in this case.