4ALinesRays


 * Lines & Rays **

Since a complex number forms a point on the complex plane, an infinite series of points, connected by an equation, can form a line.


 * A **__line__** continues to infinity in both directions.
 * A **__ray__** starts at a point and continues to infinity in one direction.
 * A **__line segment__** is a straight line joining two points.

Horizontal Line
 * {z: Im(z) = 3****}[[image:bhs-specialist/04Aline1.JPG align="right"]]**
 * **Im(z) = 3**
 * forms a horizontal line passing through **3i**

This horizontal line can then be translated vertically:
 * **{z: Im(z – 2i) = 3}**
 * shifts the line to pass through **5i**

Translating the line horizontally:
 * **{z: Im(z + 4) = 3}**
 * has no observable effect.

Vertical Line
 * {z: Re(z) = 4}**
 * forms a vertical line passing through **4**

Oblique (sloping) Line


 * An oblique line can be understood by replacing
 * **Re(z) = x** and
 * **Im(z) = y**


 * {z: 2Re(z) + 4Im(z) = 4}**
 * is therefore equivalent to the Cartesian Equation:
 * **{(x,y}: 2x + 4y = 4}**

Rays
 * {z: Arg(z) =** p**/4}**
 * forms a ray that starts at the origin and
 * extends through the first quadrant
 * at an angle of p **/4** to the positive real axis.

Note:
 * Since **Arg(0 + 0i)** is undefined,
 * there should be an open circle at the origin.
 * The ray is only defined for **r > 0**.


 * {z: Arg(z – (a + bi)) =** p**/4}**
 * translates the beginning of the ray to **a+bi**.
 * The ray maintains the same direction
 * An angle of p **/4** to the positive real axis.


 * Example **

math \text{Sketch } \; \Big\{ z: \text{Arg} \big( z -1-2i \big) = \dfrac{\pi}{6} \Big\} math

__**Solution:**__

The equation can be rewritten as: math . \qquad \Big\{ z: \text{Arg} \big( z -(1+2i) \big) = \dfrac{\pi}{6} \Big\} math

math \text{This is a ray at an angle of } \dfrac{\pi}{6} = 30^\circ \text{ to the positive real axis} math

The ray is then translated 1 unit to the right and 2 units up. .