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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2:10 minutes

Problem 61

Textbook Question

In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v.v = -10i + 15j

Verified step by step guidance

Identify the components of the vector \( \mathbf{v} = -10\mathbf{i} + 15\mathbf{j} \), where \( a = -10 \) and \( b = 15 \).

Calculate the magnitude of the vector \( ||\mathbf{v}|| \) using the formula \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \).

Substitute the values of \( a \) and \( b \) into the magnitude formula: \( ||\mathbf{v}|| = \sqrt{(-10)^2 + 15^2} \).

Determine the direction angle \( \theta \) using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

Substitute the values of \( a \) and \( b \) into the direction angle formula: \( \theta = \tan^{-1}\left(\frac{15}{-10}\right) \), and adjust the angle to the correct quadrant.

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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 and is calculated using the formula ||v|| = √(x² + y²), where x and y are the components of the vector. In this case, for the vector v = -10i + 15j, the magnitude can be found by substituting -10 for x and 15 for y, resulting in ||v|| = √((-10)² + (15)²).

Direction Angle of a Vector

The direction angle θ of a vector is the angle formed between the vector and the positive x-axis, measured counterclockwise. It can be calculated using the tangent function: θ = arctan(y/x). For the vector v = -10i + 15j, you would compute θ using the components -10 and 15, ensuring to consider the correct quadrant for the angle based on the signs of the components.

Quadrants in the Cartesian Plane

The Cartesian plane is divided into four quadrants, which are determined by the signs of the x and y coordinates. The first quadrant has both coordinates positive, the second has a negative x and positive y, the third has both negative, and the fourth has a positive x and negative y. Understanding which quadrant the vector lies in is crucial for accurately determining the direction angle θ, as it affects the angle's final value.