Questions: Plot the complex number. Then write the complex number in polar form. Express the argument in degrees. 14-14 i Plot the complex number on the complex plane to the right.

Plot the complex number. Then write the complex number in polar form. Express the argument in degrees.

14-14 i

Plot the complex number on the complex plane to the right.

Transcript text: Part 1 of 2 Points: 0 of 1 Save Plot the complex number. Then write the complex number in polar form. Express the argument in degrees. \[ 14-14 i \] Plot the complex number on the complex plane to the right.

Solution

Solution Steps

Step 1: Identify the real and imaginary parts

The complex number is given as 14 - 14i. The real part is 14 and the imaginary part is -14.

Step 2: Plot the complex number

On the complex plane, the real part is plotted on the horizontal (x) axis, and the imaginary part is plotted on the vertical (y) axis. Thus, the complex number 14 - 14i is plotted at the point (14, -14). This point is correctly shown in the provided image.

Step 3: Convert to polar form

To convert to polar form, we need to find the modulus (r) and argument (θ) of the complex number.

$r = \sqrt{14^2 + (-14)^2} = \sqrt{196 + 196} = \sqrt{392} = 14\sqrt{2}$

$\theta = \arctan\left(\frac{-14}{14}\right) = \arctan(-1) = -45^\circ$ Since the complex number is in the fourth quadrant, the argument is $315^\circ$.

The polar form of the complex number is $14\sqrt{2}(\cos{315^\circ} + i\sin{315^\circ})$.