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Chapter 6: Problem 79

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

Short Answer

Expert verified
After completing all these steps, the final answer should be given in both polar and rectangular forms.

Step by step solution

01

Convert to Polar Form

Express each complex number in polar form using the formula \(z = r(\cos 胃 + i \sin 胃)\), where \(r = |z| = \sqrt{a^2 + b^2}\) is the magnitude and \(胃 = \operatorname{atan2}(b, a)\). Here, the complex numbers are \(1 + i \sqrt{3}\) and \(2 \sqrt{3} - 2i\).
02

Perform the Operation

After converting to polar form, perform the operation, which is division in this case. For two complex numbers \(Z_1 = r_1(\cos 胃_1 + i \sin 胃_1)\) and \(Z_2 = r_2(\cos 胃_2 + i \sin 胃_2)\), the division is defined as \(Z_1/Z_2 = (r_1 / r_2)(\cos (胃_1- 胃_2) + i \sin (胃_1- 胃_2))\).
03

Convert the Result to Rectangular Form

Convert the resulted number back to rectangular form using the formula \(a = r \cos 胃\) and \(b = r \sin 胃\), where \(a\) is the real part and \(b\) is the imaginary part of the number.

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Most popular questions from this chapter

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}$$

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}+\mathbf{w})$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.

A force of 80 pounds on a rope is used to pull a box up a ramp inclined at \(10^{\circ}\) from the horizontal. The rope forms an angle of \(33^{\circ}\) with the horizontal. How much work is done pulling the box 25 feet along the ramp?

A wagon is pulled along level ground by exerting a force of 40 pounds on a handle that makes an angle of \(32^{\circ}\) with the horizontal. How much work is done pulling the wagon 100 feet? Round to the nearest foot-pound.
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