Chapter 6, Problem 5PS

a.

To determine

The solution of the value of expression.

Answer to Problem 5PS

  $\begin{array}{l}\Vert u\Vert \text{=}\sqrt{2}\\ \Vert v\Vert \text{=}\sqrt{5}\\ \Vert u+v\Vert =\text{1}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \Vert \frac{u+v}{\Vert u+v\Vert }\Vert =1\end{array}$

Explanation of Solution

Given information:

Find the value of each expression.

  $\begin{array}{l}(i)\Vert u\Vert \text{ (}ii)\Vert v\Vert \text{ (}iii)\Vert u+v\Vert \\ (iv)\Vert \frac{u}{\Vert u\Vert }\Vert \text{ (}v)\Vert \frac{v}{\Vert v\Vert }\Vert \text{ (}vi)\Vert \frac{u+v}{\Vert u+v\Vert }\Vert \end{array}$

  $(a)$

  $\begin{array}{l}u=(1,-1)\\ v=(-1,2)\end{array}$

Calculation:

From the given information, find the value of each expression by the value $u=(1,-1)$ and $v=(-1.2)$ as shown below:

  $u=(1,-1);v(-1,2);(u+v)=(0,1)$

Now plug the value in the following given expression:

  $\begin{array}{l}\Vert u\Vert \text{=}\sqrt{{(1)}^2+{(-1)}^2}\text{=}\sqrt{2}\\ \Vert v\Vert \text{=}\sqrt{{(-1)}^2+{(2)}^2}\text{=}\sqrt{5}\\ \Vert u+v\Vert =\sqrt{{(0)}^2+{(1)}^2}\text{=}\sqrt{1}\text{=1}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \Vert \frac{u+v}{\Vert u+v\Vert }\Vert =1\end{array}$

Therefore, above values are the solutions of the given expressions.

b.

To determine

The solution of the value of expression.

Answer to Problem 5PS

  $\begin{array}{l}\Vert u\Vert \text{=1 }\\ \Vert v\Vert \text{=3}\sqrt{2}\\ \Vert u+v\Vert =\sqrt{13}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \frac{\Vert u+v\Vert }{\Vert u+v\Vert }=1\end{array}$

Explanation of Solution

Given information:

Find the value of each expression.

  $\begin{array}{l}(i)\Vert u\Vert \text{ (}ii)\Vert v\Vert \text{ (}iii)\Vert u+v\Vert \\ (iv)\Vert \frac{u}{\Vert u\Vert }\Vert \text{ (}v)\Vert \frac{v}{\Vert v\Vert }\Vert \text{ (}vi)\Vert \frac{u+v}{\Vert u+v\Vert }\Vert \end{array}$

  $(b)$

  $\begin{array}{l}u=(0,1)\\ v=(3,-3)\end{array}$

Calculation:

From the given information, find the value of each expression by the value $u=(0,1)$ and $v=(3,-3)$ as shown below:

  $u=(0,1);v(3,-3);(u+v)=(3,-2)$

Now plug the value in the following given expression:

  $\begin{array}{l}\Vert u\Vert \text{=}\sqrt{{(0)}^2+{(1)}^2}\text{=}\sqrt{1}\text{=1 }\\ \Vert v\Vert \text{=}\sqrt{{(3)}^2+{(-3)}^2}\text{=}\sqrt{18}\text{=3}\sqrt{2}\\ \Vert u+v\Vert =\sqrt{{(3)}^2+{(-2)}^2}\text{=}\sqrt{13}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \frac{\Vert u+v\Vert }{\Vert u+v\Vert }=1\end{array}$

Therefore, above values are the solutions of the given expressions.

c.

To determine

The solution of the value of expression.

Answer to Problem 5PS

  $\begin{array}{l}\Vert u\Vert \text{=}\frac{\sqrt{5}}{2}\\ \Vert v\Vert \text{=}\sqrt{13}\\ \Vert u+v\Vert =\sqrt{\frac{85}{2}}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \frac{\Vert u+v\Vert }{\Vert u+v\Vert }=1\end{array}$

Explanation of Solution

Given information:

Find the value of each expression.

  $\begin{array}{l}(i)\Vert u\Vert \text{ (}ii)\Vert v\Vert \text{ (}iii)\Vert u+v\Vert \\ (iv)\Vert \frac{u}{\Vert u\Vert }\Vert \text{ (}v)\Vert \frac{v}{\Vert v\Vert }\Vert \text{ (}vi)\Vert \frac{u+v}{\Vert u+v\Vert }\Vert \end{array}$

  $(c)$

  $\begin{array}{l}u=(1,\frac{1}{2})\\ v=(2,3)\end{array}$

Calculation:

From the given information, find the value of each expression by the value $u=(1,\frac{1}{2})$ and $v=(2,3)$ as shown below:

  $u=(1,\frac{1}{2});v(2,3);(u+v)=(3,3\frac{1}{2})$

Now plug the value in the following given expression:

  $\begin{array}{l}\Vert u\Vert \text{=}\sqrt{{(1)}^2+{(\frac{1}{2})}^2}\text{=}\sqrt{\frac{5}{4}}\text{=}\frac{\sqrt{5}}{2}\\ \Vert v\Vert \text{=}\sqrt{{(2)}^2+{(3)}^2}\text{=}\sqrt{13}\\ \Vert u+v\Vert =\sqrt{{(\frac{6}{2})}^2+{(\frac{7}{2})}^2}\text{=}\sqrt{\frac{85}{2}}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \frac{\Vert u+v\Vert }{\Vert u+v\Vert }=1\end{array}$

Therefore, above values are the solutions of the given expressions.

d.

To determine

The solution of the value of expression.

Answer to Problem 5PS

  $\begin{array}{l}\Vert u\Vert \text{=}2\sqrt{5}\\ \Vert v\Vert \text{=5}\sqrt{2}\\ \Vert u+v\Vert =5\sqrt{2}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \frac{\Vert u+v\Vert }{\Vert u+v\Vert }=1\end{array}$

Explanation of Solution

Given information:

Find the value of each expression.

  $\begin{array}{l}(i)\Vert u\Vert \text{ (}ii)\Vert v\Vert \text{ (}iii)\Vert u+v\Vert \\ (iv)\Vert \frac{u}{\Vert u\Vert }\Vert \text{ (}v)\Vert \frac{v}{\Vert v\Vert }\Vert \text{ (}vi)\Vert \frac{u+v}{\Vert u+v\Vert }\Vert \end{array}$

  $(d)$

  $\begin{array}{l}u=(2,-4)\\ v=(5,5)\end{array}$

Calculation:

From the given information, find the value of each expression by the value $u=(2,-4)$ and $v=(5,5)$ as shown below:

  $u=(2,-4);v(5,5);(u+v)=(7,1)$

Now plug the value in the following given expression:

  $\begin{array}{l}\Vert u\Vert \text{=}\sqrt{{(2)}^2+{(-4)}^2}\text{=}\sqrt{20}\text{=}2\sqrt{5}\\ \Vert v\Vert \text{=}\sqrt{{(5)}^2+{(5)}^2}\text{=}\sqrt{50}\text{=5}\sqrt{2}\\ \Vert u+v\Vert =\sqrt{{(7)}^2+{(1)}^2}\text{=}\sqrt{50}=5\sqrt{2}\end{array}$

  $\begin{array}{l}\frac{\Vert u\Vert }{\Vert u\Vert }\text{=1 }\\ \frac{\Vert v\Vert }{\Vert v\Vert }\text{=1 }\\ \frac{\Vert u+v\Vert }{\Vert u+v\Vert }=1\end{array}$

Therefore, above values are the solutions of the given expressions.