## Scalars and Vectors

2.1 Scalars and Vectors

### Learning Objectives

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

- Describe the difference between vector and scalar quantities.
- Identify the magnitude and direction of a vector.
- Explain the effect of multiplying a vector quantity by a scalar.
- Describe how one-dimensional vector quantities are added or subtracted.
- Explain the geometric construction for the addition or subtraction of vectors in a plane.
- Distinguish between a vector equation and a scalar equation.

As the video explains, to describe the position of an object with respect to an origin requires that you tell not just how **FAR** from the origin it is, but **WHICH WAY**.

You need a vector to describe this. We think of it as a kind of arrow, and indicate it either just with bold letters, like * V*, or with a little arrow on top: . We will often label a vector in a diagram by its

*magnitude*, and its

*angle*with respect to some axis.

Mathematically, we write = the magnitude of the vector . (Note, the magnitude isn’t written bold and has no arrow on top.) The magnitude of any vector is always a positive number.

Examples of **vectors** in nature: velocity (has a magnitude AND direction!), force, acceleration,… I find it easy to visualize that displacement (a change in position) is a vector, but I find it a little unusual to think of position itself as a vector. But it is. Think of it as the vector which points from the origin to where the object is. We usually label position as . (Unlike the displacement vector, the position vector DOES depend on your choice of axes – your coordinate system. Maybe that’s why I find it a little unusual!?)

There are also lots of quantities in nature that are not vectors. For example, mass, speed, and time have no spatial direction associated with them. They are **scalars**.

### Defining Vectors

Vectors represent physical quantities specified completely by giving a number of units (magnitude) and a direction. A vector quantity **ALWAYS** has these two parts: * magnitude* and

*.*

**direction**When you draw a vector, the length of the vector represents the magnitude — a longer vector has a greater magnitude. The direction of the vector is denoted by the arrowhead. The arrowhead is also called the “tip” of the vector, while the other end is called the “tail”.

### Vector Notation

### Adding Vectors in One Dimension

**Discuss!**

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

A scuba diver makes a slow descent into the depths of the ocean. His vertical position with respect to a boat on the surface changes several times. He makes the first stop 9.0 m from the boat but has a problem with equalizing the pressure, so he ascends 3.0 m and then continues descending for another 12.0 m to the second stop. From there, he ascends 4 m and then descends for 18.0 m, ascends again for 7 m and descends again for 24.0 m, where he makes a stop, waiting for his buddy.

(a) Assuming the positive direction is up to the surface, express his net vertical displacement vector in terms of the unit vector.

(b) What is his distance to the boat?

### Adding Vectors in Two Dimensions

**Practice!**

Practice 2.1.1 |
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Three vectors, , , and are shown. Which of the vectors below is equal to the sum of these three? (i.e. + + = ?) |

(a) |

(b) |

(c) |

(d) |

(e) None of the above |

Practice 2.1.2 |
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Three vectors, , , and are shown. Which of the vectors below is equal to + – ? |

(a) |

(b) |

(c) |

(d) |

(e) None of the above |