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Precalculus / LEVEL 3 · DIFFICULTY 3/5

Polar, Complex and Vector Coordinates

Use coordinate systems suited to the problem.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST PC 3.1Polar CoordinatesRead the lesson
MINI QUEST PC 3.2Complex Powers and RotationsRead the lesson
MINI QUEST PC 3.3Dot ProductsRead the lesson
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STAGE PC 3.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Polar Coordinates

Useful preparation: The Cosine Rule

Goal: Understand and apply polar coordinates.

Before you begin: The Cosine Rule

ONE ROTATION, TWO COORDINATES

From the unit circle to a wave

Parent relation: $\displaystyle x^{2} + y^{2} = 1$ . The radius is 1. Turn the radius and follow the horizontal cosine and vertical sine coordinates.

Trace one complete rotation

$\displaystyle \theta = 0$ rad; $\displaystyle \operatorname{cos}\, \theta = 1$ ; $\displaystyle \operatorname{sin}\, \theta = 0$

The solid purple curve records $\displaystyle \operatorname{sin}\, \theta$ as the angle increases from 0 to $\displaystyle 2\, \pi$ .

The dashed orange curve records $\displaystyle \operatorname{cos}\, \theta$ . Both repeat after one complete turn.

Understand the idea

Polar coordinates separate distance from direction. The nonnegative radius is the distance to the origin; the angle records the point’s direction and may have many equivalent representations.

\displaystyle r^{2} = x^{2} + y^{2}

;

\displaystyle x = r\, \operatorname{cos}\, \theta

\displaystyle y = r\, \operatorname{sin}\, \theta

Choose and carry out a method

Compute $\displaystyle \sqrt{x^{2} + y^{2}}$ for the radius. When converting an angle too, use coordinate signs to select the correct quadrant.

Check the reasoning

Points opposite each other have the same radius but different directions. Negative Cartesian coordinates never force a negative distance.

WORKED EXAMPLE 1

A point has Cartesian coordinates (-8,-6). Find its nonnegative polar radius.

The nonnegative polar radius is distance from the origin.
$\displaystyle r = \sqrt{8^{2} + 6^{2}} = \sqrt{100}$ .
$\displaystyle r = 10$ . Negative coordinates affect the angle, not this radius.

WORKED EXAMPLE 2

A point has Cartesian coordinates (-15,-8). Find its nonnegative polar radius.

The nonnegative polar radius is distance from the origin.
$\displaystyle r = \sqrt{15^{2} + 8^{2}} = \sqrt{289}$ .
$\displaystyle r = 17$ . Negative coordinates affect the angle, not this radius.

Common pitfalls

Possible mix-up: A point in quadrant III has a negative radius.

The requested nonnegative radius is a distance.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why do angles differing by a full turn describe the same polar point?

Preview the eight practice prompts

Problem 1
A point has Cartesian coordinates (-35,-12). Find its nonnegative polar radius.
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Problem 2
A point has Cartesian coordinates (-48,-14). Find its nonnegative polar radius.
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Problem 3
A point has Cartesian coordinates (-63,-16). Find its nonnegative polar radius.
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Problem 4
A point has Cartesian coordinates (-80,-18). Find its nonnegative polar radius.
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Problem 5
A point has Cartesian coordinates (-99,-20). Find its nonnegative polar radius.
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Problem 6
A point has Cartesian coordinates (-120,-22). Find its nonnegative polar radius.
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Problem 7New context
A radar target is 143 units west and 24 units south of the sensor. Find the distance used in its polar coordinates.
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Problem 8New context
A radar target is 168 units west and 26 units south of the sensor. Find the distance used in its polar coordinates.
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Open stage PC 3.1 in the student workspace →

STAGE PC 3.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Complex Powers and Rotations

Useful preparation: Polar Coordinates

Goal: Understand and apply complex powers and rotations.

Before you begin: Polar Coordinates

ONE ROTATION, TWO COORDINATES

From the unit circle to a wave

Parent relation: $\displaystyle x^{2} + y^{2} = 1$ . The radius is 1. Turn the radius and follow the horizontal cosine and vertical sine coordinates.

Trace one complete rotation

$\displaystyle \theta = 0$ rad; $\displaystyle \operatorname{cos}\, \theta = 1$ ; $\displaystyle \operatorname{sin}\, \theta = 0$

The solid purple curve records $\displaystyle \operatorname{sin}\, \theta$ as the angle increases from 0 to $\displaystyle 2\, \pi$ .

The dashed orange curve records $\displaystyle \operatorname{cos}\, \theta$ . Both repeat after one complete turn.

Understand the idea

Multiplying by i rotates a complex number counterclockwise by $\displaystyle 90^{\circ}$ . Four such rotations return to the original point, giving the repeating powers 1, i, −1, −i.

\displaystyle i^{4} = 1

;

\displaystyle i^{4\, k + r} = i^{r}

Choose and carry out a method

Reduce the exponent modulo four, multiply by any outside real coefficient, and extract the requested component.

Check the reasoning

Distinguish $\displaystyle c \cdot i^{n}$ from (ci)^n: the latter also raises c to a power. A purely imaginary number has real part zero.

WORKED EXAMPLE 1

Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 3\, i^{4}$ .

Powers of i repeat every four steps: 1, i, -1, -i.
4 mod $\displaystyle 4 = 0$ ; reduce the exponent using $\displaystyle i^{4} = 1$ .
The real part is 3. Imaginary-only positions have real part zero.

WORKED EXAMPLE 2

Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 4\, i^{5}$ .

Powers of i repeat every four steps: 1, i, -1, -i.
5 mod $\displaystyle 4 = 1$ ; reduce the exponent using $\displaystyle i^{4} = 1$ .
The real part is 0. Imaginary-only positions have real part zero.

Common pitfalls

Possible mix-up: The outside coefficient is always raised to n.

Only factors inside the exponent’s parentheses are raised.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Trace the four powers of i on the complex plane.

Preview the eight practice prompts

Problem 1
Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 6\, i^{7}$ .
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Problem 2
Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 7\, i^{8}$ .
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Problem 3
Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 8\, i^{9}$ .
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Problem 4
Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 9\, i^{10}$ .
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Problem 5
Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 10\, i^{11}$ .
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Problem 6
Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 11\, i^{12}$ .
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Problem 7New context
A complex signal starts at the real value 12; each quarter-turn multiplies it by i. After 13 quarter-turns, what is its real component?
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Problem 8New context
A complex signal starts at the real value 13; each quarter-turn multiplies it by i. After 14 quarter-turns, what is its real component?
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Open stage PC 3.2 in the student workspace →

STAGE PC 3.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Dot Products

Useful preparation: Complex Powers and Rotations

Goal: Understand and apply dot products.

Before you begin: Complex Powers and Rotations

Understand the idea

The dot product combines corresponding components into a scalar. Geometrically it measures how much one vector points along the other, scaled by both lengths.

\displaystyle u \cdot v = u_{x}\, v_{x} + u

ᵧvᵧ

\displaystyle +u_{z}\, v_{z}

Choose and carry out a method

Multiply corresponding components and add all products. In a work problem, pair force and displacement in the same coordinate system.

Check the reasoning

A perpendicular pair has dot product zero. Negative work is possible when the force opposes displacement.

WORKED EXAMPLE 1

Find the dot product of $\displaystyle u = \left(3, 3, -2\right)$ and $\displaystyle v = \left(4, 2, 5\right)$ .

A dot product pairs corresponding components and adds their products.
$\displaystyle u \cdot v = 3 \cdot 4 + 3 \cdot 2 + \left(-2\right) \cdot 5$ .
The scalar result is 8. Negative terms represent opposing components.

WORKED EXAMPLE 2

Find the dot product of $\displaystyle u = \left(4, 3, -2\right)$ and $\displaystyle v = \left(4, 3, 5\right)$ .

A dot product pairs corresponding components and adds their products.
$\displaystyle u \cdot v = 4 \cdot 4 + 3 \cdot 3 + \left(-2\right) \cdot 5$ .
The scalar result is 15. Negative terms represent opposing components.

Common pitfalls

Possible mix-up: A dot product returns a vector.

It returns a scalar measuring directional alignment.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Explain why a force perpendicular to motion performs zero work in this model.

Preview the eight practice prompts

Problem 1
Find the dot product of $\displaystyle u = \left(6, 3, -2\right)$ and $\displaystyle v = \left(4, 5, 5\right)$ .
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Problem 2
Find the dot product of $\displaystyle u = \left(7, 3, -2\right)$ and $\displaystyle v = \left(4, 6, 5\right)$ .
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Problem 3
Find the dot product of $\displaystyle u = \left(8, 3, -2\right)$ and $\displaystyle v = \left(4, 7, 5\right)$ .
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Problem 4
Find the dot product of $\displaystyle u = \left(9, 3, -2\right)$ and $\displaystyle v = \left(4, 8, 5\right)$ .
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Problem 5
Find the dot product of $\displaystyle u = \left(10, 3, -2\right)$ and $\displaystyle v = \left(4, 9, 5\right)$ .
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Problem 6
Find the dot product of $\displaystyle u = \left(11, 3, -2\right)$ and $\displaystyle v = \left(4, 10, 5\right)$ .
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Problem 7New context
A constant force (12,3,-2) N moves an object by (4,11,5) m. Find the work in joules.
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Problem 8New context
A constant force (13,3,-2) N moves an object by (4,12,5) m. Find the work in joules.
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Open stage PC 3.3 in the student workspace →

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Polar, Complex and Vector Coordinates

Polar Coordinates

From the unit circle to a wave

Understand the idea

Choose and carry out a method

Check the reasoning

A point has Cartesian coordinates (-8,-6). Find its nonnegative polar radius.

A point has Cartesian coordinates (-15,-8). Find its nonnegative polar radius.

Common pitfalls

Explain it to yourself

Complex Powers and Rotations

From the unit circle to a wave

Understand the idea

Choose and carry out a method

Check the reasoning

Let i2=−1\displaystyle i^{2} = -1i2=−1. Find the real part of 3 i4\displaystyle 3\, i^{4}3i4.

Let i2=−1\displaystyle i^{2} = -1i2=−1. Find the real part of 4 i5\displaystyle 4\, i^{5}4i5.

Common pitfalls

Explain it to yourself

Dot Products

Understand the idea

Choose and carry out a method

Check the reasoning

Find the dot product of u=(3,3,−2)\displaystyle u = \left(3, 3, -2\right)u=(3,3,−2) and v=(4,2,5)\displaystyle v = \left(4, 2, 5\right)v=(4,2,5).

Find the dot product of u=(4,3,−2)\displaystyle u = \left(4, 3, -2\right)u=(4,3,−2) and v=(4,3,5)\displaystyle v = \left(4, 3, 5\right)v=(4,3,5).

Common pitfalls

Explain it to yourself

Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 3\, i^{4}$ .

Let $\displaystyle i^{2} = -1$ . Find the real part of $\displaystyle 4\, i^{5}$ .

Find the dot product of $\displaystyle u = \left(3, 3, -2\right)$ and $\displaystyle v = \left(4, 2, 5\right)$ .

Find the dot product of $\displaystyle u = \left(4, 3, -2\right)$ and $\displaystyle v = \left(4, 3, 5\right)$ .