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Chapter 6: Problem 80

Convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$\frac{(-1+i \sqrt{3})(2-2 i \sqrt{3})}{4 \sqrt{3}-4 i}$$

Short Answer

Expert verified
Performing these steps will give the results both in polar and rectangular form of the complex number as required by the exercise.

Step by step solution

01

Conversion to Polar Form

Convert the complex numbers in the numerator and denominator to their polar forms. The formula used to convert rectangular to polar form is \(r = \sqrt{{a^2 + b^2}}\) where \(a\) and \(b\) are the real and imaginary parts of the complex number respectively. Also, the angle \(\theta\) can be obtained by \(\tan^{-1} (\frac{b}{a})\). Using these formulas, the complex numbers found on the numerator are converted to their polar forms and denoted as \(Z_1\) and \(Z_2\) respectively. Similarly, the complex number in denominator converts to \(Z_3\). Calculate the polar forms.
02

Perform Operations

Perform the mathematical operations as indicated in the exercise. Since the division operation is represented in polar form as \(\frac{Z_1 \times Z_2}{Z_3}\), this operation involves multiplying the magnitudes and subtracting the angles.
03

Convert Back to Rectangular Form

Finally, convert the results back into rectangular form using the conversion formula \(a = r \cos(\theta)\) for real part and \(b = r \sin(\theta)\) for imaginary part. Compute to get the rectangular form.

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Using words and no symbols, describe how to find the \(\mathrm{d}\) product of two vectors with the alternative formula $$\mathbf{v} \cdot \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| \cos \theta$$

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