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 7.47

Textbook Question

Write each vector in the form a i + b j.
〈2, 0〉

Verified step by step guidance

Identify the components of the vector 〈2, 0〉. Here, the first component is 2 and the second component is 0.

Understand that the vector 〈2, 0〉 can be expressed in terms of the unit vectors **i** and **j**.

The unit vector **i** represents the x-direction, and the unit vector **j** represents the y-direction.

Multiply the first component (2) by the unit vector **i** to get the x-component of the vector: 2i.

Multiply the second component (0) by the unit vector **j** to get the y-component of the vector: 0j.

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Key Concepts

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

Vector Representation

Vectors can be represented in a Cartesian coordinate system using unit vectors. The standard form of a vector in two dimensions is expressed as 'a i + b j', where 'a' is the component along the x-axis (i) and 'b' is the component along the y-axis (j). This notation allows for easy manipulation and understanding of vector quantities.

Components of a Vector

The components of a vector indicate its magnitude in each direction of the coordinate system. For the vector 〈2, 0〉, the first component '2' represents the horizontal (x-axis) movement, while the second component '0' indicates no movement in the vertical (y-axis) direction. Understanding these components is crucial for converting vectors into their standard form.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction. In the context of vector representation, 'i' and 'j' are unit vectors that point in the positive x and y directions, respectively. They serve as the building blocks for expressing any vector in the plane, allowing for clear communication of direction and magnitude.

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CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.

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Textbook Question

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Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈5, 7〉