Exam 8: Polar Coordinates, Vectors, and Complex Numbers

Sketch the graph for the polar equation for .

Use the dot product to find the angle between the vectors (-2, -4) and (4, 5). Express your answer in radians to four decimal places.

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

Convert the rectangular coordinates (-19, 19) to polar coordinates (r, $\theta$ ). Use radians, and always choose the angle to lie in the interval (- $\pi$ , $\pi$ ]. Give the exact answer.

Find two distinct numbers k such that . Give the exact answers.

Evaluate

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

Find the center and radius of the circle whose equation in polar coordinates is r =12sin $\theta$ .

Write the expression in the form a + bi, where a and b are real numbers. (8 + 9i)²

Suppose u = (3, -5) and v = (4, -3). Find u - v.

Find an angle that determines the direction of the vector (-2, 5). Express your answer in decimal radians rounded to three decimal places.

Sketch the graph for the polar equation for .

Find two complex numbers whose sum equals 14 and whose product equals 58.

Sketch the graph for the polar equation for .

If u and v are two vectors with the same initial point, then $\left| u-v \right|$ equals the distance between the endpoint of u and the endpoint of v.

Find two possible complex solution to the equation. Give the exact answers.

Write the expression in the form a + bi, where a and b are real numbers. (8 + 7i) - (4 + 10i)

Write the expression in the form a + bi, where a and b are real numbers. (10 + 4i)(6 - 4i)

Find three distinct complex numbers z such that z³ = -2

Let z = 10-2i and w = 9-2i be two complex numbers. The imaginary part of z + w is -4i.