Question

In Exercises 40 through 46, consider vectors $\mathit{v}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{,}\mathit{v}{\mathbf{⃗}}_{\mathbf{2}}\mathbf{,}\mathit{v}{\mathbf{⃗}}_{\mathbf{3}}$in ${\mathit{R}}^{\mathbf{4}}$; we are told that $\mathit{v}{\mathbf{⃗}}_{\mathbf{i}}\mathbf{,}\mathit{v}{\mathbf{⃗}}_{\mathbf{j}}$is the entry ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$of matrix A.

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$

Find a nonzero vector $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$in span $\left({\stackrel{\rightharpoonup }{v}}_2{\stackrel{\rightharpoonup }{v}}_3\right)$such that $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$is orthogonal to ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}}_{\mathbf{3}}$.Express as a linear combination of localid="1659441496004" ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}}_{\mathbf{2}}$and ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}}_{\mathbf{3}}$.

Short answer

${\stackrel{\rightharpoonup }{v}}_3$The required projection is,$\stackrel{\rightharpoonup }{v}={\stackrel{\rightharpoonup }{v}}_2-pro{j}_{\nabla 3}{\stackrel{\rightharpoonup }{v}}_2$is orthogonal to .

Step-by-step solution

Formula for the orthogonal projection

If Vis a subspace of ${\mathit{R}}^{\mathbf{n}}$with an orthonormal basis$\mathit{u}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{u}{\mathbf{⃗}}_{\mathbf{m}}$then

$\mathit{p}\mathit{r}\mathit{o}{\mathit{j}}_{\mathbf{v}}\mathit{x}\mathbf{⃗}\mathbf{=}{\mathit{x}}^{\mathbf{I}\mathbf{I}}\mathbf{⃗}\mathbf{}\mathbf{=}\mathbf{(}\mathit{u}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{.}\mathit{x}\mathbf{⃗}\mathbf{)}\mathit{u}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\mathbf{(}\mathit{u}{\mathbf{⃗}}_{\mathbf{m}}\mathbf{.}\mathit{x}\mathbf{⃗}\mathbf{)}\mathit{u}{\mathbf{⃗}}_{\mathbf{m}}$

For $x⃗$all in $R^n.$

Let us write the matrix in the$v{⃗}_i.v{⃗}_j$notation.

Consider the terms below.

$$\begin{gathered} A=\left|3511 \\ 5920 \\ 112049\right|=\left[{\stackrel{\rightharpoonup }{V}}_1.{\stackrel{\rightharpoonup }{V}}_1{\stackrel{\rightharpoonup }{V}}_1.{\stackrel{\rightharpoonup }{V}}_2{\stackrel{\rightharpoonup }{V}}_1.{\stackrel{\rightharpoonup }{V}}_3 \\ {\stackrel{\rightharpoonup }{V}}_2.{\stackrel{\rightharpoonup }{V}}_1{\stackrel{\rightharpoonup }{V}}_2.{\stackrel{\rightharpoonup }{V}}_2{\stackrel{\rightharpoonup }{V}}_2.{\stackrel{\rightharpoonup }{V}}_3 \\ {\stackrel{\rightharpoonup }{V}}_3.{\stackrel{\rightharpoonup }{V}}_1{\stackrel{\rightharpoonup }{V}}_3.{\stackrel{\rightharpoonup }{V}}_2{\stackrel{\rightharpoonup }{V}}_3.{\stackrel{\rightharpoonup }{V}}_3\right]=\left[{\left|\left|{\stackrel{\rightharpoonup }{V}}_1\right|\right|}^2{\stackrel{\rightharpoonup }{V}}_1.{\stackrel{\rightharpoonup }{V}}_2{\stackrel{\rightharpoonup }{V}}_1.{\stackrel{\rightharpoonup }{V}}_3 \\ {\stackrel{\rightharpoonup }{V}}_2.{\stackrel{\rightharpoonup }{V}}_1{\left|\left|{\stackrel{\rightharpoonup }{V}}_2\right|\right|}^2{\stackrel{\rightharpoonup }{V}}_2.{\stackrel{\rightharpoonup }{V}}_3 \\ {\stackrel{\rightharpoonup }{V}}_3.{\stackrel{\rightharpoonup }{V}}_1{\stackrel{\rightharpoonup }{V}}_3.{\stackrel{\rightharpoonup }{V}}_2{\left|\left|{\stackrel{\rightharpoonup }{V}}_3\right|\right|}^2\right] \end{gathered}$$

Consider the vector $v⃗=v{⃗}_2-pro{j}_{}\left(v{⃗}_3\right)v{⃗}_2$

Now, according to the above theorem an orthonormal basis of span$v{⃗}_3$, ,which is

$u⃗=\frac{v{⃗}_3}{\left|\right|v{⃗}_3\left|\right|}=\frac{v{⃗}_3}{7}$

Determine that v⃗=v⃗2-2049v⃗3 is orthogonal to or not

To find out that the above $\overrightarrow{V}=\overrightarrow{V_z}-\frac{20}{{49}_{}}{\stackrel{\rightharpoonup }{v}}_3$term is orthogonal to or not consider the equations below.

$$\begin{gathered} \stackrel{\rightharpoonup }{v}.{\stackrel{\rightharpoonup }{v}}_3={\stackrel{\rightharpoonup }{v}}_2-\frac{20}{49}{\stackrel{\rightharpoonup }{v}}_3 \\ ={\stackrel{\rightharpoonup }{v}}_2.{\stackrel{\rightharpoonup }{v}}_3-\frac{20}{49}{\stackrel{\rightharpoonup }{.v}}_3.{\stackrel{\rightharpoonup }{v}}_3 \\ =20-\frac{20}{49}\times 49 \\ =0 \end{gathered}$$

Thus,${\stackrel{\rightharpoonup }{v}}_{}$ is orthogonal to ${\stackrel{\rightharpoonup }{v}}_3$.

Hence,$\stackrel{\rightharpoonup }{v}={\stackrel{\rightharpoonup }{v}}_2-pro{j}_{v_3}{\stackrel{\rightharpoonup }{v}}_2$ is orthogonal to ${\stackrel{\rightharpoonup }{v}}_3$.