Chapter 8, (js), Polar Coordinates and Vectors

Plane flying N @300mph has vector v; 40mph N30°E wind has vector u. Plane's true speed and direction will be the sum (w) of those vectors. What are the vectors u and v, and what is the magnitude and direction of w.

u = 0i + 300j. v = 40cos60(i) + 40sin60(j) = 20i + 34.64j. lwl = sqrt 20^2 + 334.64^2. The direction of w is arctan 334.64/20.

8.5 The cos of the angle between two vectors = ....

.... their dot product divided by the product of their magnitudes. cosθ = u·v/lul·lvl

Write the polar or trigonometric form of the complex number -4sqrt3 - 4i.

8(cos 7pi/6 + isin 7pi/6). Note, signs in parentheses are always (+); cos and sin of the angles will give you the correct signs.

If the r coordinate is negative in the polar coordinate system, what does that mean?

A point P(r, theta) lies r units from the pole in the direction opposite to that given by theta.

Where are the tips of the lemniscates?

At "a," so take sqrt of a^2.

How do you subtract the 2 vectors, u and v, graphically?

Draw u and v with same starting point. Draw the parallels to u and v to complete the parallelogram, but draw the parallel to v in the opposite direction of v since it is u - v. ie, give the neg sign to v. Then draw the other diagonal of the parallelogram.

8.3 - Polar Form of Complex Numbers; DeMoivre's Theorem How do you graph a complex number such as 7 + 3i?

Graph them using the complex plane, which has the real axis along the horizontal, and the imaginary axis vertical. 7 + 3i would be 7 over and then 3 up.

When are 2 vectors equal?

If they have same magnitude and direction.

How would you graph the set of all complex numbers Z such that the abs values, or moduli, lZl of those complex numbers is less than or equal to 1?

It would be a circle of radius 1, that is filled in to include all points w/in the circle.

8.4 - Vectors Length and mass are _________ . The book also refers to real numbers as _________. Quantities that involve magnitude and direction are ________ _________.

Scalars, scalars, directed quantities.

How are the locations of the points P(5, pi/4) and Q(-5, 5pi/4) different? A point in the plane can be represented by how many polar coordinates?

They aren't. Infinitely many, because P(r, θ) can be represented by P(r, θ + 2npi) and P(-r, θ + (2n + 1)pi).

Directed quantities can be represented mathematically through the use of _______.

Vectors

What are you adding and subtracting when adding and subtracting vectors?

You are adding and subtracting their hztl and vertical components, not coordinates in a plane.

8.2 - Graphs of Polar Equations Sketch the graphs of r = 3, and θ = pi/4, and express the equations in rectangular coordinates.

js588

Sketch the graphs of r = 4cos(θ), and r = -4sin(θ)

js589

The Dot Product Theorem states that the dot product of 2 vectors = ? (hint: it involves θ)

lul·lvl·cosθ (ie, the dot product of 2 vectors equals the product of their magnitudes times the cos of the angle between them.)

The length of a vector is its:

magnitude, denoted by the vector inside vertical lines, eg, lABl with an arrow over AB.

What is the equation for a lemniscate? How do you graph lemniscate on ti84?

r^2 = a^2(cos or sin)(2θ). You have to take sqrt of both sides, and then enter it as 2 equations (the + and - roots of the R hand side of the equation).

8.2 Find the polar coordinates for the point w/ rectangular coordinates of (2, -2). jsp pg584

r^2 = x^2 + y^2, so r can be (+/-)2sqrt2. tanθ = -2/2, tan^-1 = -pi/4. Although tan^-1 is restricted to (-pi/2, pi/2), giving the -pi/4 value, 3pi/4 is also a solution for the angle coordinate. Since the point is in quadrant 4, the answers are (2sqrt2, -pi/4), and (-2sqrt2, 3pi/4).

What is the position for the sin and cos lemniscates?

sin lemniscates at 45/135, and cos lemniscates at 0/180.

What are the formulas for changing from rectangular to polar coordinates?

x^2 + y^2 = r^2, y/x = tan θ

Write the polar form of the complex number z = a + bi.

z = r(cos θ + isin θ), where r = lzl = sqrt a^2 + b^2, and tan θ = b/a. r is the modulus of z, and θ is the argument of z. r here is just like the r used as the first coordinate in the polar coordinate system. A point P in the polar coordinate system has coordinates (r, θ). The polar form of the complex number z = a + bi = r(cos θ + isin θ). These are 2 different things.

DeMoivre's Theorem is the formula for raising a complex number to a power of n. Write it.

z^n = r^n[cos (n*theta) + isin (n*theta)]

8.1 - Polar Coordinates What are the polar equivalents of the origin, the (+) x-axis, and coordinates in the rectangular coordinate system?

The pole, the polar axis, and (r, theta).

There are 3 tests for symmetry for polar equations; I don't think I need them if I use JG's method for graphing. A positive result for the tests is conclusive for symmetry, but a negative result is inconclusive; i.e., may still be symmetry. There are other tests for symmetry besides the following 3.

1. If polar equation unchanged when θ replaced by (-θ), graph is symmetric about polar axis. 2. If polar equation unchanged when θ replaced by (pi + θ), graph is symmetric about the pole. Remember the to use the addition formula for sin/cos. 3. If polar equation unchanged when θ replaced by (pi - θ), graph is symmetric about the vertical line θ = pi/2. Remember the to use the subtraction formula for sin/cos.

Multiplication of a vector by a scalar can stretch, shrink, or change the direction of a vector. What is the effect if the scalar is 0.5, 2, and -1.

Shrink, stretch, change direction.

How do you add vectors u and v graphically?

Put initial point of v at terminal point of u, then draw vector u+v connecting initial point of u to terminal point of v. Or, draw vectors u and v with same starting point, and then the diagonal of the parallelogram is the sum u+v. js608

How many arguments can a complex number have?

There can be more than one argument of a complex number, and any two arguments differ by a multiple of 2 pi.

8.5 - The Dot Product What is the dot product of vectors u and v, where u = <a1, b1>, and v = <a2, b2>? u*u = ? NOTE: The dot product is a scalar; it is NOT a vector.

The dot product u·v = a1·a2 + b1·b2. lul^2 (ie, u*u equals the magnitude of u squared)

How do you find the nth roots of complex numbers? How do you graph the nth roots of complex numbers? (An aside: What is the 8th root of 16? 16 raised to the 1/4th, and that quantity raised to the 1/2. So, sqrt of 2. Sqrt of 2 raised to the 8th power = 16.

See the Box on page 601. The nth roots of the complex number z will be equally spaced on the circle of radius r^1/n in the complex plane.

What is the modulus of a complex number?

The modulus (or abs value) of the complex number "z" in the complex plane is its distance from the origin. lzl = sqrt a^2 + b^2. e.g., lzl = l3 + 4il = sqrt 3^2 + 4^2 = sqrt 25 = 5. For z = 8 - 5i, lzl = l8 - 5il = sqrt 8^2 + (-5)^2 = sqrt 89

What are the formulas for changing from polar coordinates to the x and y rectangular coordinates?

x = r(cos θ), y = r(sin θ)