## Displacement and Velocity Vectors

4.1 Displacement and Velocity Vectors

### Learning Objectives

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

- Calculate position vectors in a multidimensional displacement problem.
- Solve for the displacement in two or three dimensions.
- Calculate the velocity vector given the position vector as a function of time.
- Calculate the average velocity in multiple dimensions.

### Position and Displacement Vectors in Three Dimensions

The **position vector** from the origin of the coordinate system to a point is

If the particle is moving, the variables *x*, *y*, and *z* are functions of time (*t*): *x*(*t*), *y*(*t*), and *z*(*t*).

The **displacement vector** is found by subtracting from :

### Velocity and Average Velocity in Three Dimensions

In the previous chapter we found the instantaneous velocity by calculating the derivative of the position function with respect to time. We can do the same operation in two and three dimensions, but we use vectors. The **instantaneous** **velocity vector** is

We can also write the instantaneous velocity vector in vector notation:

where , , and .

**Practice!**

Practice 4.1.1 |
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Which of these is a key difference between one-dimensional (1D) and two-dimensional (2D) vectors? |

(a) 1D vectors have a direction indicated by either a (+) or (-) sign, but 2D vectors have a more complex direction. |

(b) 1D vectors only have a magnitude while 2D vectors have both magnitude and direction. |

(c) 1D vectors are always positive, but 2D vectors can be positive or negative. |

(d) 1D vectors only have one possible direction, but 2D vectors have two possible directions. |

Practice 4.1.2 |
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Consider a two-dimensional (2D) velocity vector in the x–y plane, . The x-component of ______. |

(a) Can only be found using , regardless of how is measured. |

(b) Can only be found using , regardless of how is measured. |

(c) Can be found using either or , depending on how is measured. |

Practice 4.1.3 |
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As an object moves in the x–y plane, which statement is true about the object’s instantaneous velocity at a given moment? |

(a) The instantaneous velocity is perpendicular to the object’s path. |

(b) The instantaneous velocity is tangent to the object’s path. |

(c) The instantaneous velocity can point in any direction, independent of the object’s path. |

Practice 4.1.4 |
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As an object moves in the x–y plane, which statement is true about the object’s average velocity? |

(a) The average velocity for some time period equals the object’s position at the end of that period divided by the time at the end of that period. |

(b) The average velocity for some time period equals the object’s displacement during that period divided by that period of time. |

(c) The average velocity is equal to the object’s current position divided by the current time. |

(d) The average velocity for some time period is equal to the period of time divided by the object’s displacement during that period. |

**Discuss!**

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

The position of a particle is

(a) Determine its velocity and acceleration as functions of time.

(b) What are its velocity and acceleration at time *t* = 0?