Exam 8: Polar Coordinates, Vectors, and Complex Numbers

Find three distinct numbers z such that . Give the exact answer.

Write the expression in the form a + bi, where a and b are real numbers. (-7 + 3i)(5 - 2i)

Find the center and radius of the circle whose equation in polar coordinates is r = 9cos $\theta$ . Give the exact answers.

Evaluate .

Find two real numbers a such that = 5.

Evaluate . Give the exact answer.

Suppose u = (3, -6). Find -7u.

Suppose u = (8, -1). Evaluate $|\mathbf{u}|$

Write 7 in polar form.

Convert the rectangular coordinates (-5, 2) to polar coordinates (r, $\theta$ ). Use radians, and always choose the angle to lie in the interval (- $\pi$ , $\pi$ ]. Round your answers to the nearest thousandth.

Write the expression in the form a + bi, where a and b are real numbers. (3 + 6i) - (2 - 5i)

Suppose the wind at airplane heights is 40 miles per hour (relative to the ground) moving 18° south of east. An airplane wants to fly directly north at 340 miles per hour relative to the ground. Find the speed and direction that the airplane must fly relative to the wind. Round the speed to 2 decimal places and the direction to a tenth of a degree.

Convert the rectangular coordinates (0, -7) to polar coordinates (r, $\theta$ ). Use radians, and always choose the angle to lie in the interval (- $\pi$ , $\pi$ ]. Give the exact answer.

Write in polar form. Give the exact answer using radians for .

Convert the polar coordinates r = 3, to rectangular coordinates in the xy-plane. Give the exact answer.

Find the complex conjugate of .

Find four distinct complex numbers z such that z⁴ = -5. Give the exact answers.

Which of the following complex numbers satisfy the equation z³ = 7i?

Write the expression in the form a + bi, where a and b are real numbers. (2 + 7i)(9 + 9i)

Write the expression in the form a + bi, where a and b are real numbers. (4 + 9i) - (3 - 10i)