Question

In Exercise 40 through 46, consider vectors ${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{3}}$in role="math" localid="1659606624550" ${\mathit{R}}^{\mathbf{4}}$; we are told that ${\overrightarrow{\mathbf{v}}}_{\mathbf{i}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{j}}$is the entry data-custom-editor="chemistry" ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$of matrix A.

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

Find $\mathbf{\left|}\mathbf{\right|}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{\left|}\mathbf{\right|}$

Short answer

The value of$\left|\right|{\overrightarrow{v}}_1+{\overrightarrow{v}}_2\left|\right|$ is $\approx 4.69$.

Step-by-step solution

Step by step answerStep 1: Given

The given matrix is $\mathrm{A}=\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$.

Determine the value of ||v→1+v→2||

To find the value of $\left|\right|{\overrightarrow{v}}_1+{\overrightarrow{v}}_2\left|\right|$. Let us write the matrix in the ${\overrightarrow{v}}_i.{\overrightarrow{v}}_j$notation.

Consider the terms below:

$\mathrm{A}=\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]=\left[\begin{array}{ccc}{\overrightarrow{v}}_1.{\overrightarrow{v}}_1& {\overrightarrow{v}}_1.{\overrightarrow{v}}_2& {\overrightarrow{v}}_1.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_2.{\overrightarrow{v}}_1& {\overrightarrow{v}}_2.{\overrightarrow{v}}_2& {\overrightarrow{v}}_2.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_3.{\overrightarrow{v}}_1& {\overrightarrow{v}}_3.{\overrightarrow{v}}_2& {\overrightarrow{v}}_3.{\overrightarrow{v}}_3\end{array}\right]=\left[\begin{array}{ccc}\left|\right|{\overrightarrow{v}}_1|{|}^2& {\overrightarrow{v}}_1.{\overrightarrow{v}}_2& {\overrightarrow{v}}_1.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_2.{\overrightarrow{v}}_1& \left|\right|{\overrightarrow{v}}_2|{|}^2& {\overrightarrow{v}}_2.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_3.{\overrightarrow{v}}_1& {\overrightarrow{v}}_3.{\overrightarrow{v}}_2& \left|\right|{\overrightarrow{v}}_3|{|}^2\end{array}\right]$

Now, compute the above terms in the form of $\left|\right|{\overrightarrow{v}}_1+{\overrightarrow{v}}_2\left|\right|$:

$$\begin{gathered} \left|\right|{\overrightarrow{v}}_1+{\overrightarrow{v}}_2|{|}^2=({\overrightarrow{v}}_1+{\overrightarrow{v}}_2)·({\overrightarrow{v}}_1+{\overrightarrow{v}}_2) \\ ={\overrightarrow{v}}_1.{\overrightarrow{v}}_1+{\overrightarrow{v}}_1.{\overrightarrow{v}}_2+{\overrightarrow{v}}_2.{\overrightarrow{v}}_1+{\overrightarrow{v}}_2.{\overrightarrow{v}}_2 \\ =3+5+5+9 \\ =22 \\ ||{\overrightarrow{v}}_1+{\overrightarrow{v}}_2||=\sqrt{22} \\ \approx 4.69 \end{gathered}$$

Thus,

$\left|\right|{\overrightarrow{v}}_1+{\overrightarrow{v}}_2\left|\right|\approx 4.69$