Question

Consider the orthonormal vectors ${\overrightarrow{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}{\overrightarrow{\mathbf{u}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{u}}}_{\mathbf{m}}$in ${\mathit{R}}^{\mathbf{n}}$, and an arbitrary vector $\overrightarrow{\mathbf{x}}$in ${\mathit{R}}^{\mathbf{n}}$what is the relationship between the following two quantities $\mathit{p}\mathbf{=}{\left({\overrightarrow{u}}_1.\overrightarrow{x}\right)}^{\mathbf{2}}\mathbf{+}{\left({\overrightarrow{u}}_2.\overrightarrow{x}\right)}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\left({\overrightarrow{u}}_m.\overrightarrow{x}\right)}^{\mathbf{2}}$and role="math" localid="1659541405973" ${||\overrightarrow{x}||}^{\mathbf{2}}$When are the two quantities equal?

Short answer

$p⩽m\left|\right|\overrightarrow{x}|{|}^2$and they are equal if and only if $\overrightarrow{x}$lies in V.

Step-by-step solution

Step by step solution Step 1: Use Cauchy-Schwarz inequality

$\mathbf{|}\overrightarrow{\mathbf{v}}\mathbf{.}\overrightarrow{\mathbf{w}}\mathbf{|}\mathbf{⩽}\mathbf{\left|}\mathbf{\right|}\overrightarrow{\mathbf{v}}\mathbf{\left|}\mathbf{\right|}\mathbf{.}\mathbf{\left|}\mathbf{\right|}\overrightarrow{\mathbf{w}}\mathbf{\left|}\mathbf{\right|}$

Firstly we will apply the Pythagorean Theorem and by applying it we get

${}_{i=1}{}^m{\left({\overrightarrow{u}}_1.\overrightarrow{x}\right)}^2\left|\right|u_i|{|}^2={}_{i=1}{}^m{\left({\overrightarrow{u}}_1.\overrightarrow{x}\right)}^2=p$

Now, apply the Cauchy-Schwarz inequality.

$$\begin{gathered} |{\overrightarrow{u}}_1.\overrightarrow{x}|⩽\left|\right|u_i\left|\right|·\left|\right|\overrightarrow{x}\left|\right|=\left|\right|\overrightarrow{x}\left|\right| \\ \Rightarrow {\left({\overrightarrow{u}}_1.\overrightarrow{x}\right)}^2⩽\left|\right|\overrightarrow{x}|{|}^2 \\ \Rightarrow {}_{i=1}{}^m{\left({\overrightarrow{u}}_1.\overrightarrow{x}\right)}^2=p⩽{}_{i=1}{}^m||\overrightarrow{x}|{|}^2=m|\left|\overrightarrow{x}\right|{|}^2 \end{gathered}$$

 Step 2: Compare the quantities

Consider the diagram below.

Hence, $p⩽m\left|\right|\overrightarrow{x}|{|}^2$and they will be equal if and only if lies $\overrightarrow{x}$in V.