Question

Find the Magnitude and the angle of the complex number A given below:

A= (3+j4)(-3+j4) / (-3-j4)

Answer 1

Answer 2

Given Complex Number:

$$A=\frac{(3+j4)(-3+j4)}{-3-j4}$$

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Step 1: Multiply the Numerator

First, compute the product $(3+j4)(-3+j4)$ using the distributive property (FOIL method):

$$(3+j4)(-3+j4)=3\cdot (-3)+3\cdot j4+j4\cdot (-3)+j4\cdot j4$$$$=-9+j12-j12+j^{2}16$$$$=-9+j(12-12)+(-1)\cdot 16(\text{since }{j}^2=-1)$$$$=-9-16=-25$$

The numerator simplifies to $-25$.

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Step 2: Divide by the Denominator

Now, divide the numerator by the denominator $(-3-j4)$:

$$A=\frac{-25}{-3-j4}=\frac{25}{3+j4}$$

To simplify, multiply the numerator and denominator by the complex conjugate of the denominator $(3-j4)$:

$$A=\frac{25(3-j4)}{(3+j4)(3-j4)}$$$$=\frac{75-j100}{9-j^{2}16}$$$$=\frac{75-j100}{9+16}(\text{since }{j}^2=-1)$$$$=\frac{75-j100}{25}$$$$=3-j4$$

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Step 3: Compute Magnitude and Angle

The simplified form of $A$ is $3-j4$.

1. Magnitude ($\mathrm{\mid }A\mathrm{\mid }$):

   $$\mathrm{\mid }A\mathrm{\mid }=\sqrt{3^2+(-4{)}^2}=\sqrt{9+16}=\sqrt{25}=5$$

2. Angle ($\theta$):
   The angle is calculated using $\theta ={\tan }^{-1}\left(\frac{\text{Imaginary}}{\text{Real}}\right)$:

   $$\theta ={\tan }^{-1}\left(\frac{-4}{3}\right)\approx -53.13^{\circ }$$

   Since the complex number is in the fourth quadrant (positive real, negative imaginary), the angle is:

   $$\theta =360^{\circ }-53.13^{\circ }=306.87^{\circ }(\text{or }-53.13^{\circ })$$

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Answer:

• Magnitude of $A$: $\boxed{{\displaystyle 5}}$

• Angle of $A$: $\boxed{{\displaystyle -53.13^{\circ }}}$ or $\boxed{{\displaystyle 306.87^{\circ }}}$