Chapter 9.3, Problem 44E

To determine

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

$2$ Units to the right and down $3$ units

Given:

The given operation is $w=z+(2-3i)$

Calculation:

The transformation $w=z+(2-3i)$ translate $2$ units to the right and down $3$ units because it adds $2$ the x component and $-3$ to the y component.

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

  ${90}^{\circ }$ Counter clockwise.

Given:

The given operation is $w=i\cdot z$

Calculation:

The transformation $w=i\cdot z$ rotates $z$ , ${90}^{\circ }$ counter clockwise.

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

Dilation by a factor of $3$ .

Given:

The given operation is $w=3z$

Calculation:

The transformation $w=3z$ dilates $z$ by a factor of $3$ .

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

Reflection of $z$ about the real axis.

Given:

The given operation- $w$ is the conjugate of $z$

Calculation:

Reflect $z$ about the real axis because change the y component to $-y$ .

To find the rectangular form of these vectors.

Answer to Problem 44E

$2$ Units to the right and down $3$ units

Explanation of Solution

Given:

The given operation is $w=z+(2-3i)$

Calculation:

The transformation $w=z+(2-3i)$ translate $2$ units to the right and down $3$ units because it adds $2$ the x component and $-3$ to the y component.

To determine

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

Answer to Problem 44E

$2$ Units to the right and down $3$ units

Explanation of Solution

Given:

The given operation is $w=z+(2-3i)$

Calculation:

The transformation $w=z+(2-3i)$ translate $2$ units to the right and down $3$ units because it adds $2$ the x component and $-3$ to the y component.

To determine

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

Answer to Problem 44E

  ${90}^{\circ }$ Counter clockwise.

Explanation of Solution

Given:

The given operation is $w=i\cdot z$

Calculation:

The transformation $w=i\cdot z$ rotates $z$ , ${90}^{\circ }$ counter clockwise.

To determine

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

Answer to Problem 44E

Dilation by a factor of $3$ .

Explanation of Solution

Given:

The given operation is $w=3z$

Calculation:

The transformation $w=3z$ dilates $z$ by a factor of $3$ .

To determine

To describe the transformation applied to point $z$ to obtain point $w$ in the complex plane for the given operation.

Answer to Problem 44E

Reflection of $z$ about the real axis.

Explanation of Solution

Given:

The given operation- $w$ is the conjugate of $z$

Calculation:

Reflect $z$ about the real axis because change the y component to $-y$ .

To determine

To add the two vectors found in part (a).

Answer to Problem 44E

  $(5.47,6.07)$

Explanation of Solution

Given:

The phasors for these two expressions are written as $4\angle {20}^{\circ }$ and $5\angle {70}^{\circ }$

Calculation:

Adding the two vectors as follows-

  $(3.76,1.37)+(1.71,4.70)=(5.47,6.07)$ .

To determine

To write the sum of two vectors in phasor notation.

Answer to Problem 44E

  $8.17\angle {47.98}^{\circ }$

Explanation of Solution

Given:

The phasors for these two expressions are written as $4\angle {20}^{\circ }$ and $5\angle {70}^{\circ }$

Calculation:

It is given that

  $r\cos \theta =5.47,r\sin \theta =6.07$ .

Now it can be concluded that-

  $\begin{array}{l}\frac{r\sin \theta }{r\cos \theta }=\frac{6.07}{5.47}\\ \tan \theta =\frac{6.07}{5.47}\\ \theta =\arctan \left(\frac{6.07}{5.47}\right)\\ \theta ={47.98}^{\circ }\\ r=\frac{5.47}{\cos {47.98}^{\circ }}=8.17\end{array}$

Therefore the phasor notation is $8.17\angle {47.98}^{\circ }$

To determine

To write the sinusoidal expression corresponding to the phasor in part c.

Answer to Problem 44E

  $r\sin (3.14t+\theta )=8.17\sin (3.14t+47.98)$

Explanation of Solution

Given:

The phasors for these two expressions are written as $4\angle {20}^{\circ }$ and $5\angle {70}^{\circ }$

Calculation:

The sinusoidal expression corresponding to that phasor in part c is-

  $r\sin (3.14t+\theta )=8.17\sin (3.14t+47.98)$ .