Question

Consider a subspace Vof ${\mathit{R}}^{\mathbf{n}}$and a vector $\overrightarrow{\mathbf{x}}$ in${\mathit{R}}^{\mathbf{n}}$. Let $\overrightarrow{\mathbf{y}}\mathbf{=}\mathit{p}\mathit{r}\mathit{o}{\mathbf{j}}_{\mathbf{V}}\overrightarrow{\mathbf{x}}$. What is the relationship between the following quantities?

$\mathbf{\left|}\mathbf{\right|}\overrightarrow{\mathbf{y}}\mathbf{|}{\mathbf{|}}^{\mathbf{2}}$and $\overrightarrow{\mathbf{y}}\mathbf{.}\overrightarrow{\mathbf{x}}$

Short answer

The relationship is$\left|\right|\overrightarrow{y}|{|}^2=\overrightarrow{y}·\overrightarrow{x}$,

Step-by-step solution

Step by step solution Step 1: Draw the diagram

Draw the diagram which represent the given condition as:

Determine the relationship

Let us first recall the geometric meaning of $\overrightarrow{y}=\text{pro}{\text{j}}_V\overrightarrow{x}$. From the above diagram, it is clear that $\overrightarrow{y}-\overrightarrow{x}$ is perpendicular to $\overrightarrow{y}$and it is denoted by $x^{\left|\right|}=\overrightarrow{x}-{\overrightarrow{x}}^{\perp }$.

Thus, we have:

$$\begin{gathered} \overrightarrow{y}\overrightarrow{x}\perp \overrightarrow{y} \\ \Rightarrow \left(\overrightarrow{y}-\overrightarrow{x}\right)·\overrightarrow{y}=0 \\ \Rightarrow \overrightarrow{y}.\overrightarrow{y}-\overrightarrow{x}·\overrightarrow{y}=0 \\ \Rightarrow \left|\right|\overrightarrow{y}|{|}^2-\overrightarrow{x}·\overrightarrow{y}=0 \\ \Rightarrow |\left|\overrightarrow{y}\right|{|}^2=\overrightarrow{x}·\overrightarrow{y}=\overrightarrow{y}·\overrightarrow{x} \end{gathered}$$

Hence, Relation between the $\left|\right|\overrightarrow{y}|{|}^2$ and $\overrightarrow{y}·\overrightarrow{x}$ is $\left|\right|\overrightarrow{y}|{|}^2=\overrightarrow{y}·\overrightarrow{x}$.