## Coordinate Systems and Components of a Vector

2.2 Coordinate Systems and Components of a Vector

### Learning Objectives

By the end of this section, you will be able to:

- Describe vectors in two and three dimensions in terms of their components, using unit vectors along the axes.
- Distinguish between the vector components of a vector and the scalar components of a vector.
- Explain how the magnitude of a vector is defined in terms of the components of a vector.
- Identify the direction angle of a vector in a plane.
- Explain the connection between polar coordinates and Cartesian coordinates in a plane.

### Trig Review

Before we get into the discussion about vector components, let’s first do a quick review of trigonometry. Test yourself with the following practice problems.

**Practice!**

Practice 2.2.1 |
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From the figure, how do you define the ? |

(a) |

(b) |

(c) |

(d) |

(e) |

Practice 2.2.2 |
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From the figure, how do you define the ? |

(a) |

(b) |

(c) |

(d) |

(e) |

Practice 2.2.3 |
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From the figure, how do you define the ? |

(a) |

(b) |

(c) |

(d) |

(e) |

Practice 2.2.4 |
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What is the length of the side of the triangle adjacent to Î¸ if Î¸ = 30Â° and the hypotenuse is 2 units? |

(a) |

(b) |

(c) |

(d) |

(e) |

Practice 2.2.5 |
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What is the length of the side of the triangle opposite to Î¸ if Î¸ = 30Â° and the hypotenuse is 2 units? |

(a) |

(b) |

(c) |

(d) |

(e) |

Practice 2.2.6 |
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From the triangle in the figure, what is another way to express c? |

(a) |

(b) |

(c) |

(d) |

(e) |

### Vector Components

**Practice!**

Practice 2.2.7 |
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What is the magnitude of the x-component of vector ? |

(a) 14.4 m |

(b) 10.8 m |

(c) 13.6 m |

(d) 18.0 m |

(e) 12.0 m |

Practice 2.2.8 |
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What is the magnitude of the y-component of vector ? |

(a) 14.4 m |

(b) 10.8 m |

(c) 13.6 m |

(d) 18.0 m |

(e) 12.0 m |

### Adding Vectors using Vector Components

**Practice!**

Practice 2.2.9 |
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When vector is added to vector , the resultant is vector such that . = + . What is the magnitude of the x-component of the resultant vector ? |

(a) 26.4 m |

(b) 10.8 m |

(c) 22.8 m |

(d) 6.00 m |

(e) 2.40 m |

Practice 2.2.10 |
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When vector is added to vector , the resultant is vector such that . = + . What is the magnitude of the y-component of the resultant vector ? |

(a) 26.4 m |

(b) 10.8 m |

(c) 22.8 m |

(d) 6.00 m |

(e) 2.40 m |

Practice 2.2.11 |
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When vector is added to vector , the resultant is vector such that . = + . This resultant vector has components and . How do you determine the magnitude of ? |

(a) |

(b) |

(c) |

(d) |

(e) |

Practice 2.2.12 |
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When vector is added to vector , the resultant is vector such that . = + . This resultant vector has components and . How do you determine the direction of , the angle the vector makes with the positive x-axis? |

(a) |

(b) |

(c) |

(d) |

(e) |

Practice 2.2.13 |
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Vectors and have equal magnitudes but point in different directions, as shown in the figure. What is true about the xÂ-component of vector compared to the x-component of vector ? |

(a) is equal in magnitude and points in the same direction to . |

(b) is greater in magnitude and points in the opposite direction to . |

(c) is greater in magnitude and points in the same direction to . |

(d) is equal in magnitude and points in the opposite direction to . |

Practice 2.2.14 |
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Vectors and have equal magnitudes but point in different directions, as shown in the figure. What is true about the y-component of vector compared to the y-component of vector ? |

(a) is equal in magnitude and points in the same direction to . |

(b) is greater in magnitude and points in the opposite direction to . |

(c) is greater in magnitude and points in the same direction to . |

(d) is equal in magnitude and points in the opposite direction to . |

Practice 2.2.15 |
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Vectors and have equal magnitudes but point in different directions, as shown in the figure. If you add + , what is true about the resultant vector = + ? |

(a) The resultant points in the positive y-direction and has a magnitude equal to A + B. |

(b) The resultant points in the positive x-direction and has a magnitude less than A + B. |

(c) The resultant points in the positive x-direction and has a magnitude equal to A + B. |

(d) The resultant points in the positive y-direction and has a magnitude less than A + B. |

(e) The resultant points in the negative x-direction and has a magnitude less than A + B. |

**Discuss!**

Consider how you would answer this question. Then bring this to class for a group discussion.

An ant crawling on a table undergoes three consecutive displacements:

Find the magnitude and direction of the resultant displacement.