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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Problem 7.27c

Textbook Question

Use the figure to find each vector: - u. Use vector notation as in Example 4.

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Verified step by step guidance

Identify the given vector \( \mathbf{u} \) from the figure. Note its direction and magnitude.

Understand that finding \(-\mathbf{u}\) involves reversing the direction of \(\mathbf{u}\) while keeping the same magnitude.

If \(\mathbf{u} = \langle a, b \rangle\), then \(-\mathbf{u} = \langle -a, -b \rangle\).

Apply the concept of vector negation to the components of \(\mathbf{u}\) as identified from the figure.

Express the resulting vector \(-\mathbf{u}\) in the same vector notation as \(\mathbf{u}\).

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Key Concepts

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

Vector Notation

Vector notation is a mathematical representation of vectors, typically expressed in the form of ordered pairs or triples, such as u = <x, y> in two dimensions or u = <x, y, z> in three dimensions. This notation allows for clear communication of a vector's direction and magnitude, which is essential for performing vector operations like addition, subtraction, and scalar multiplication.

Magnitude of a Vector

The magnitude of a vector is a measure of its length, calculated using the Pythagorean theorem. For a vector u = <x, y>, the magnitude is given by ||u|| = √(x² + y²). Understanding how to compute the magnitude is crucial for analyzing vectors, especially when determining their relative sizes or when normalizing them to unit vectors.

Vector Components

Vector components refer to the projections of a vector along the axes of a coordinate system. For example, in a two-dimensional space, a vector u can be broken down into its horizontal (x) and vertical (y) components. This decomposition is vital for solving problems involving vectors, as it simplifies calculations and helps in visualizing vector addition and other operations.