Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

8. Vectors

Geometric Vectors

Trigonometry

8. Vectors

Geometric Vectors

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1:54 minutes

Problem 2

Textbook Question

In Exercises 1–4, u and v have the same direction. In each exercise:Find ||v||.

Verified step by step guidance

insert step 1> Identify that vectors $ \mathbf{u} $ and $ \mathbf{v} $ have the same direction, meaning they are scalar multiples of each other.

insert step 2> Express $ \mathbf{v} $ as $ \mathbf{v} = k \mathbf{u} $, where $ k $ is a scalar.

insert step 3> Use the property of magnitudes: $ ||\mathbf{v}|| = |k| \cdot ||\mathbf{u}|| $.

insert step 4> Determine the scalar $ k $ by comparing the components of $ \mathbf{u} $ and $ \mathbf{v} $ if given, or use any additional information provided in the problem.

insert step 5> Calculate $ ||\mathbf{v}|| $ using the formula from step 3, substituting the known values of $ k $ and $ ||\mathbf{u}|| $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnitude of a Vector

The magnitude of a vector, denoted as ||v||, represents its length or size in a given direction. It is calculated using the formula ||v|| = √(v₁² + v₂² + ... + vₙ²) for a vector v = (v₁, v₂, ..., vₙ) in n-dimensional space. Understanding how to compute the magnitude is essential for analyzing vectors in trigonometry.

Direction of Vectors

Vectors have both magnitude and direction, and when two vectors u and v have the same direction, they are scalar multiples of each other. This means that one vector can be expressed as a scaled version of the other, which is crucial for solving problems involving vector relationships. Recognizing this property helps in simplifying calculations and understanding vector operations.

Unit Vectors

A unit vector is a vector with a magnitude of 1, used to indicate direction without regard to magnitude. To find a unit vector in the direction of a vector v, you divide v by its magnitude: u = v / ||v||. This concept is important in trigonometry as it allows for the normalization of vectors, making it easier to work with directional components in various applications.