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The figure shows a small plane flying at a speed of 180 miles per hour on a bearing of \(\mathrm{N} 50^{\circ} \mathrm{E}\). The wind is blowing from west to east at 40 miles per hour. The figure indicates that \(\mathbf{v}\) represents the velocity of the plane in still air and w represents the velocity of the wind. a. Express \(v\) and \(w\) in terms of their magnitudes and direction angles. b. Find the resultant vector, \(\mathbf{v}+\mathbf{w}\) c. The magnitude of \(v+w\), called the ground speed of the plane, gives its speed relative to the ground. Approximate the ground speed to the nearest mile per hour. d. The direction angle of \(v+w\) gives the plane's true course relative to the ground. Approximate the true course to the nearest tenth of a degree. What is the plane's true bearing?

A plane is flying at a speed of 320 miles per hour on a bearing of \(\mathrm{N} 70^{\circ} \mathrm{E}\). Its ground speed is 370 miles per hour and its true course is \(30^{\circ} .\) Find the speed, to the nearest mile per hour, and the direction angle, to the nearest tenth of a degree, of the wind.

In calculus, it can be shown that $$e^{i \theta}=\cos \theta+i \sin \theta$$ In Exercises \(87-90,\) use this result to plot each complex number. $$ e^{\frac{\pi i}{6}} $$

Exercises \(116-118\) will help you prepare for the material covered in the next section. Use the distance formula to determine if the line segment with endpoints \((-3,-3)\) and \((0,3)\) has the same length as the line segment with endpoints \((0,0)\) and \((3,6)\)