Problem 2: Fix an angle $\theta$, and let $f_\theta: \mathbb{C} \to \mathbb{C}$ be the mapping $$f_\theta(z) = e^{i\theta} \cdot z.$$ Show that, as a mapping, $f_\theta$ represents a rotation around the origin by an angle of $\theta$.
Added by Monica L.
Close
Step 1
Then, $f_\theta(z) = e^{i\theta} \cdot z = e^{i\theta} \cdot (r e^{i\alpha}) = r e^{i(\alpha + \theta)}$. Show more…
Show all steps
Your feedback will help us improve your experience
Ajay Singhal and 96 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Show that the multiplication of any complex number $z$ by $e^{j}$ is describable, in geometrical terms, as a positive rotation through the angle $\theta$ of the vector by which $z$ is represented, without any alteration of its length.
If we multiply a complex number $z_{1}=r_{1} e^{i \theta_{1}}$ by $e^{i \theta_{2}}$, we get $e^{1 \theta_{2}} z_{1}=r_{1} e^{\left.i \theta_{1}+\theta_{2}\right)}$, that is, a complex number with the same $r$, but with its angle increased by $\theta_{2}$. We can say that the vector $r_{1}$ from the origin to the point $z_{1}$ has been rotated by the angle $\theta_{2}$. In Figure 6.1, we rotated the axes through angle $\theta$ which is equivalent to rotating the vector $\mathrm{r}$ through the angle $-\theta$, leaving the axes unchanged. Thus if $z=x+i y$, we shouid find $x^{\prime}+i y^{\prime}=z^{\prime}=e^{-i \theta} z=e^{-t 0}(x+t y)$. Take real and imaginary parts of this equation to obtain equations $(6.3)$.
LINEAR EQUATIONS; VECTORS, MATRICES, AND DETERMINANTS
Matrix operations
Explain what is meant by saying that multiplying a complex number $z=r(\cos \theta+i \sin \theta)$ by $i$ amounts to rotating $z 90^{\circ}$ counterclockwise around the origin. [Hint: Express $i$ and $i z$ in polar form. What are their relative positions in the complex plane?]
Applications of Trigonometry
The Complex Plane and Polar Form for Complex Numbers
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD