Exam 7: Vectors, the Complex Plane, and Polar Coordinates

Find the angle (round to the nearest degree) between the vectors. < -7, -28 > and < -6, -22 >

Find the product, z₁z₂ in rectangular form. and

Use a calculator to express the complex number in rectangular form. Round to four decimal places.

Find the result of the expression using De Moivre's theorem. Write the answer in rectangular form.

Use a calculator to express the complex number in rectangular form. Round to four decimal places. -10(cos 236° + i sin 236°)

Convert the point to exact rectangular coordinates.

Find all nth roots of z. Write answers in polar form.

Use a calculator to express the complex number in polar form. Round angle to one decimal place and in degrees.

Find the quotient, In rectangular form. Z₁ = 72 [ cos 218° + i sin 218° ] and z₂ = 6 [ cos 68° + i sin 68° ]

To slide a crate across the floor, a force of 807 pounds at a 28° is needed. How much work is done if the crate is dragged 86 feet. Round to the nearest integer.

Find all nth roots of z. Write answers in polar form.

Graph the equation. r² = 4 cos 5θ

How much work does it take to lift 144 pounds vertically 4.5 feet?

Express the complex number in rectangular form. 18(cos 330° + i sin 330°)

The lemniscate motion occurs naturally in the flapping of birds' wings. The bird's vertical lift and wing sweep create the distinctive figure-eight pattern. The patterns vary with the different wing profiles. Compare the two possible lemniscate patterns by graphing them on the same polar graph: r² = cos 2θ and r² = cos (2θ + 1).

Find the magnitude and direction angle of the vector u. U = <-40, -9>

Spirals are seen in nature - in the swirl of a pine cone; they are also used in machinery to convert motions. An Archimedes Spiral has the general equation r = a θ. A more general form for the quation of a spiral is r = aθ^1/n where n is a constant that determines how tightly the spiral is wrapped. Compare the Archimedes Sprial r = θ with the spiral by graphing both on the same polar graph.

Find all nth roots of z. Write answers in polar form.

Find the quotient, in rectangular form. z₁ = 208 [ cos 180° + i sin 180° ] and z₂ = 8 [ cos 45° + i sin 45° ]

A plane has a heading of 33° and an airspeed of 540 mph. The wind is blowing at 75 mph from 123°. What are its actual heading and airspeed? Round to one decimal place.