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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2:28 minutes

Problem 39

Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.v = 6i

Verified step by step guidance

Identify the given vector \( \mathbf{v} = 6\mathbf{i} \).

Calculate the magnitude of the vector \( \mathbf{v} \). The magnitude \( ||\mathbf{v}|| \) is given by \( ||\mathbf{v}|| = \sqrt{6^2} \).

Simplify the expression for the magnitude: \( ||\mathbf{v}|| = \sqrt{36} \).

Determine the unit vector \( \mathbf{u} \) by dividing the vector \( \mathbf{v} \) by its magnitude: \( \mathbf{u} = \frac{\mathbf{v}}{||\mathbf{v}||} = \frac{6\mathbf{i}}{\sqrt{36}} \).

Simplify the expression for the unit vector: \( \mathbf{u} = \frac{6\mathbf{i}}{6} \).

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Key Concepts

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

Unit Vector

A unit vector is a vector that has a magnitude of one and indicates direction. To find a unit vector in the same direction as a given vector, you divide the vector by its magnitude. This process normalizes the vector, allowing it to retain its direction while having a standardized length.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y² + z²) for a vector in three-dimensional space. For a two-dimensional vector, it simplifies to √(x² + y²). Understanding how to compute the magnitude is essential for normalizing a vector to find its unit vector.

Vector Notation

Vector notation typically uses letters with an arrow or bold type to represent vectors, such as v = 6i, where 'i' denotes the unit vector in the x-direction. Recognizing this notation is crucial for interpreting vector components and performing operations like finding unit vectors, as it helps in visualizing the vector's direction and magnitude.