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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2:38 minutes

Problem 49

Textbook Question

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.||v|| = 12, θ = 225°

Verified step by step guidance

Start by understanding that a vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) can be expressed as \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \), where \( v_x \) and \( v_y \) are the components of the vector.

Use the given magnitude \( ||\mathbf{v}|| = 12 \) and direction angle \( \theta = 225^\circ \) to find the components \( v_x \) and \( v_y \).

Calculate \( v_x \) using the formula \( v_x = ||\mathbf{v}|| \cdot \cos(\theta) \). Substitute the given values: \( v_x = 12 \cdot \cos(225^\circ) \).

Calculate \( v_y \) using the formula \( v_y = ||\mathbf{v}|| \cdot \sin(\theta) \). Substitute the given values: \( v_y = 12 \cdot \sin(225^\circ) \).

Combine the components to express the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \): \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \).

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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 represents its length or size and is denoted by ||v||. In this case, the magnitude is given as 12, indicating that the vector extends 12 units from the origin in a specified direction. Understanding magnitude is crucial for determining how far the vector reaches in space.

Direction Angle

The direction angle θ of a vector indicates the angle formed with the positive x-axis, measured counterclockwise. Here, θ is 225°, which places the vector in the third quadrant of the Cartesian plane. This angle helps in determining the vector's orientation and is essential for converting the vector into its component form.

Vector Components

A vector can be expressed in terms of its components along the x-axis (i) and y-axis (j). The components are calculated using the formulas v_x = ||v|| * cos(θ) and v_y = ||v|| * sin(θ). For θ = 225°, these calculations will yield the specific i and j components, allowing for a complete representation of the vector in a two-dimensional space.