Table of contents

0. Review of College Algebra4h 45m

Rationalizing Denominators
17m

Pythagorean Theorem & Basics of Triangles
17m

Basics of Graphing
30m

Functions
41m

Transformations
45m

Asymptotes
4m

Solving Linear Equations
31m

Solving Quadratic Equations
55m

Complex Numbers
41m

1. Measuring Angles40m

Angles in Standard Position
12m

Coterminal Angles
8m

Complementary and Supplementary Angles
10m

Radians
8m

2. Trigonometric Functions on Right Triangles2h 5m

Trigonometric Functions on Right Triangles
45m

Special Right Triangles
30m

Cofunctions of Complementary Angles
26m

Solving Right Triangles
23m

3. Unit Circle1h 19m

Defining the Unit Circle
14m

Trigonometric Functions on the Unit Circle
9m

Common Values of Sine, Cosine, & Tangent
11m

Reference Angles
38m

Reciprocal Trigonometric Functions on the Unit Circle
6m

4. Graphing Trigonometric Functions1h 19m

Graphs of the Sine and Cosine Functions
32m

Phase Shifts
14m

Graphs of Secant and Cosecant Functions
10m

Graphs of Tangent and Cotangent Functions
21m

5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m

Inverse Sine, Cosine, & Tangent
28m

Evaluate Composite Trig Functions
44m

Linear Trigonometric Equations
28m

6. Trigonometric Identities and More Equations2h 34m

Introduction to Trigonometric Identities
1h 4m

Sum and Difference Identities
1h 3m

Double Angle Identities
16m

Solving Trigonometric Equations Using Identities
9m

7. Non-Right Triangles1h 38m

Law of Sines
49m

Law of Cosines
30m

Area of SAS & ASA Triangles
19m

8. Vectors2h 25m

Geometric Vectors
28m

Vectors in Component Form
32m

Direction of a Vector
23m

Unit Vectors and i & j Notation
27m

Dot Product
22m

Cross Product
11m

9. Polar Equations2h 5m

Polar Coordinate System
29m

Convert Points Between Polar and Rectangular Coordinates
26m

Convert Equations Between Polar and Rectangular Forms
29m

Graphing Other Common Polar Equations
39m

10. Parametric Equations1h 6m

Graphing Parametric Equations
12m

Eliminate the Parameter
32m

Writing Parametric Equations
21m

11. Graphing Complex Numbers1h 7m

Graphing Complex Numbers
6m

Polar Form of Complex Numbers
22m

Products and Quotients of Complex Numbers
14m

Powers of Complex Numbers (DeMoivre's Theorem)
23m

8. Vectors

Dot Product

Trigonometry

8. Vectors

Dot Product

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Multiple Choice

If $u$ is a unit vector, and $v$ and $w$ are also unit vectors, which of the following is always true about the dot products $u \cdot v$ and $u \cdot w$ ?

Both

u \cdot v

and

u \cdot w

are always equal to

1

Both

u \cdot v

and

u \cdot w

are always negative.

Both

u \cdot v

and

u \cdot w

are always equal to

0

Both

u \cdot v

and

u \cdot w

are between

-1

and

1

, inclusive.

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Verified step by step guidance

Recall that the dot product of two vectors $ \mathbf{a} $ and $ \mathbf{b} $ is defined as $ \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta $, where $ \theta $ is the angle between the vectors.

Since $ \mathbf{u} $, $ \mathbf{v} $, and $ \mathbf{w} $ are all unit vectors, their magnitudes are $ \|\mathbf{u}\| = \|\mathbf{v}\| = \|\mathbf{w}\| = 1 $. Substitute these into the dot product formula to get $ \mathbf{u} \cdot \mathbf{v} = \cos \theta_{uv} $ and $ \mathbf{u} \cdot \mathbf{w} = \cos \theta_{uw} $.

Understand that the cosine of any angle $ \theta $ lies between $ -1 $ and $ 1 $, inclusive. This means $ -1 \leq \cos \theta \leq 1 $.

Therefore, the dot products $ \mathbf{u} \cdot \mathbf{v} $ and $ \mathbf{u} \cdot \mathbf{w} $ must also lie between $ -1 $ and $ 1 $, inclusive, because they are equal to the cosine of the angles between the vectors.

Conclude that the only statement always true about these dot products is that both $ \mathbf{u} \cdot \mathbf{v} $ and $ \mathbf{u} \cdot \mathbf{w} $ are between $ -1 $ and $ 1 $, inclusive.