Exam 8: Polar Coordinates, Vectors, and Complex Numbers

Sketch the graph for the polar equation for .

Find an angle that determines the direction of the vector (-6, -4). Express your answer in decimal radians rounded to three decimal places.

Write the expression in the form a + bi, where a and b are real numbers. (5 - 9i)²

Convert the rectangular coordinates (-9, -1) 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.

Find a complex number z such that

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

Write the expression in the form a + bi, where a and b are real numbers. i¹¹⁸²

Use the dot product to find the angle between the vectors (3, 9) and (2, 5). Express your answer in radians to four decimal places.

Suppose u = (-1, 4) and v = (3, -1). A) Draw a figure illustrating the sum of u and v as arrows. B) Compute the sum u + v using coordinates.

Write the expression in the form a + bi, where a and b are real numbers.

Convert the polar coordinates r = 17, to rectangular coordinates in the xy-plane. Give the exact answer.

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

Evaluate .

Evaluate .

Suppose w and z are complex numbers. If the real part of wz equals the real part of w times the real part of z, then either w or z is a real number.

Write the expression in the form a + bi, where a and b are real numbers.

Write 10i in polar form. Give the exact answer using radians for .

Sketch the graph for the polar equation for .

Suppose and . Find the polar form of . Give the exact answer using radians for .

Suppose and . Find the polar form of . Give the exact answer using radians for .