Chapter 14.2, Problem 27WE

(a)

To determine

To find: the exponent form of the vector $\overrightarrow{u}$ and $\overrightarrow{v}$ .

Answer to Problem 27WE

The exponent form of the vectors are $\overrightarrow{u}=7.51\text{i}+2.73\text{j}$ and $\overrightarrow{v}=2.95\text{i}+16.74\text{j}$

Explanation of Solution

Given information:

The values are given by $\Vert u\Vert =8$ , $\Vert v\Vert =17$ , $\alpha =20°$ , and $\beta =80°$ .

The exponent form of the vector can be written as,

  $\overrightarrow{u}=\Vert u\Vert \cos \alpha +\Vert u\Vert \sin \alpha$ and $\overrightarrow{v}=\Vert v\Vert \cos \beta +\Vert v\Vert \sin \beta$ .

Substitute 8for $\Vert u\Vert$ , and $20°$ for $\alpha$ in the formula $\overrightarrow{u}=\Vert u\Vert \cos \alpha +\Vert u\Vert \sin \alpha$ to find the vector $\overrightarrow{u}$ .

  $\begin{array}{c}\overrightarrow{u}=8\cos 20+8\sin 20\\ =7.51\text{i}+2.73\text{j}\end{array}$

Substitute $17$ for $\Vert v\Vert$ , and $80°$ for $\beta$ in the formula $\overrightarrow{v}=\Vert v\Vert \cos \beta +\Vert v\Vert \sin \beta$ to find the vector $\overrightarrow{v}$ .

  $\begin{array}{c}\overrightarrow{v}=17\cos 80+17\sin 80\\ =2.95\text{i}+16.74\text{j}\end{array}$

Therefore, the exponent form of the vectors are $\overrightarrow{u}=7.51\text{i}+2.73\text{j}$ and $\overrightarrow{v}=2.95\text{i}+16.74\text{j}$ .

(b)

To determine

To find: the dot product of the vector $\overrightarrow{u}$ and $\overrightarrow{v}$ .

Answer to Problem 27WE

The dot product of the vectors is $68$ .

Explanation of Solution

Given information:

The vectors are given by $\overrightarrow{u}=7.51\text{i}+2.73\text{j}$ and $\overrightarrow{v}=2.95\text{i}+16.74\text{j}$ .

Consider the vectors.

  $\overrightarrow{u}=7.51\text{i}+2.73\text{j}$

  $\overrightarrow{v}=2.95\text{i}+16.74\text{j}$

Find the dot product of the vectors.

  $\begin{array}{c}\overrightarrow{u}·\overrightarrow{v}=\left(7.51\text{i}+2.73\text{j}\right)·\left(2.95\text{i}+16.74\text{j}\right)\\ =\left(7.5\right)·\left(2.95\right)+\left(2.73\right)\left(16.74\right)\\ \approx 68\end{array}$

Therefore, the dot product of the vectors is $68$ .

(c)

To determine

To find: the exponent form of the vector $\overrightarrow{u}$ and $\overrightarrow{v}$ .

Answer to Problem 27WE

The dot product of the vectors is $68$ .

Explanation of Solution

Given information:

The values are given by $\Vert u\Vert =8$ , $\Vert v\Vert =17$ , $\alpha =20°$ , and $\beta =80°$ .

The dot product of the two vectors can be written as,

  $\overrightarrow{u}·\overrightarrow{v}=\Vert u\Vert \Vert v\Vert \cos \theta$ .

Find the value of $\theta$ .

  $\begin{array}{c}\theta =80-20\\ =60\end{array}$

Substitute 8 for $\Vert u\Vert$ , $17$ for $\Vert v\Vert$ , and $60°$ for $\theta$ in the formula $\overrightarrow{u}=\Vert u\Vert \cos \alpha +\Vert u\Vert \sin \alpha$ to find the vector $\overrightarrow{u}$ .

  $\begin{array}{c}\overrightarrow{u}·\overrightarrow{v}=\left(8\right)\left(17\right)\cos 60°\\ =\left(57.42\right)0.0872\\ \approx 68\end{array}$

Therefore, the dot product of the vectors is $68$ .