Question

Can you find a line Lin ${\mathit{R}}^{\mathbf{n}}$and a vector$\overrightarrow{\mathbf{x}}$ in ${\mathit{R}}^{\mathbf{n}}$such that$\overrightarrow{\mathbf{x}}\mathbf{·}\mathbf{\text{pro}}{\mathbf{\text{j}}}_{\mathbf{L}}\overrightarrow{\mathbf{x}}$is negative? Explain, arguing algebraically.

Short answer

For any$\overrightarrow{x}\in R^n$ and line L inrole="math" localid="1659602164103" $R^{\mathrm{n}}$ , we have$\overrightarrow{x}·pro{j}_L\left(\overrightarrow{x}\right)⩾0$

Step-by-step solution

Step by step solution Step 1: Given:

The given condition is $\overrightarrow{x}·pro{j}_L\left(\overrightarrow{x}\right)$ is negative.

Find a line L in Rn

The answer is in the negative sense. That is, it is impossible to find such a line and a vector in $R^n$.observe that, given any line L and vector $\overrightarrow{x}\in R^n$we have:

$$\begin{gathered} \overrightarrow{x}·{\text{proj}}_L\left(\overrightarrow{x}\right)\perp {\text{proj}}_L\left(\overrightarrow{x}\right) \\ \Rightarrow \left(\overrightarrow{x}·{\text{proj}}_L\left(\overrightarrow{x}\right)\right)·{\text{proj}}_L\left(\overrightarrow{x}\right)=0 \\ \Rightarrow \overrightarrow{x}·{\text{proj}}_L\left(\overrightarrow{x}\right)-\left|\right|{\text{proj}}_L\left(\overrightarrow{x}\right)|{|}^2 \\ \Rightarrow \overrightarrow{x}·{\text{proj}}_L\left(\overrightarrow{x}\right)=|\left|{\text{proj}}_L\left(\overrightarrow{x}\right)\right||⩾0 \end{gathered}$$

Hence,

For any $\overrightarrow{x}\in R^n$ and line L in $R^n$, there is $\overrightarrow{x}·{\text{proj}}_L\left(\overrightarrow{x}\right)⩾0$.