Describe Geometrically The Sets Of Points Z In The Complex P

Question

Describe Geometrically The Sets Of Points Z In The Complex Plane Defin Analyze and interpret geometrically the sets of points $z$ in the complex plane defined by the following relations: (1) $|z - z_1| = |z - z_2|$ where $z_1, z_2 \in \mathbb{C}$. (2) $1/z = z$. (3) $\operatorname{Re}(z) = 3$. (4) $\operatorname{Re}(z) > c$, (resp., $\geq c$) where $c \in \mathbb{R}$. (5) $\operatorname{Re}(az + b) > 0$ where $a, b \in \mathbb{C}$. (6) $|z| = \operatorname{Re}(z) + 1$. (7) $\operatorname{Im}(z) = c$ with $c \in \mathbb{R}$. Additionally, describe the properties of the inner products: the usual inner product $\langle \cdot, \cdot \rangle$ in $\mathbb{R}^2$ and the Hermitian inner product $(\cdot, \cdot)$ in $\mathbb{C}$, and demonstrate that $\langle z, w \rangle = \frac{1}{2}[(z,w) + (w,z)] = \operatorname{Re}(z,w)$, where $z, w \in \mathbb{C}$, with the identification $z = x + iy \in \mathbb{C}$ corresponding to $(x, y) \in \mathbb{R}^2$.

Dr. Jack HW Helper · Accepted Answer

Understanding geometric sets in the complex plane provides profound insights into complex analysis and its applications. Each of the given relations characterizes a particular geometric locus, such as lines, circles, or regions, revealing structural properties of complex points $z = x + iy$. This paper explores these sets systematically, alongside a discussion of the properties of two inner products—one in $\mathbb{R}^2$ and a Hermitian inner product in $\mathbb{C}$—as well as their interrelation. 1. The locus of points equidistant from two points: $|z - z_1| = |z - z_2|$ This relation describes the set of points $z$ in the complex plane equidistant from two fixed points $z_1, z_2 \in \mathbb{C}$. Geometrically, this set corresponds to the perpendicular bisector of the segment joining $z_1$ and $z_2$. To see this, note that $|z - z_1| = |z - z_2|$ implies that $z$ is located symmetrically with respect to the midpoint of $z_1$ and $z_2$, forming a straight line perpendicular to the segment connecting the two points—this is a classic locus of points equidistant from two points. When expressed explicitly, if $z_k = x_k + iy_k$, the locus satisfies the linear equation obtained from the equality of Euclidean distances, resulting in a straight line in $\mathbb{R}^2$. 2. The set of points satisfying $1/z = z$ This relation simplifies to $z^2 = 1$, which yields the solutions $z = \pm 1$. Geometrically, these solutions represent two distinct points on the real axis in the complex plane. The set therefore consists of exactly two points: $-1$ and $1$. Thus, this relation describes a discrete set of points rather than a continuous locus. 3. The vertical line where $\operatorname{Re}(z) = 3$ The condition $\operatorname{Re}(z) = 3$ describes a vertical line in the complex plane at $x=3$. All points $z = x + iy$ with $x=3$, where $y$ traverses $\mathbb{R}$, belong to this set. Geometrically, it is an infinite straight line paral

Describe Geometrically The Sets Of Points Z In The Complex P

Describe Geometrically The Sets Of Points Z In The Complex Plane Defin

Paper For Above instruction

1. The locus of points equidistant from two points: \(|z - z_1| = |z - z_2|\)

2. The set of points satisfying \(1/z = z\)

3. The vertical line where \(\operatorname{Re}(z) = 3\)

4. The half-plane \(\operatorname{Re}(z) > c\) and \(\operatorname{Re}(z) \geq c\)

5. The set where \(\operatorname{Re}(az + b) > 0\) for \(a, b \in \mathbb{C}\)

6. The relation \(|z| = \operatorname{Re}(z) + 1\)

7. The line \(\operatorname{Im}(z) = c\)

Inner products in \(\mathbb{R}^2\) and \(\mathbb{C}\)

References