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Vectors: Theories and Principles

Here we will examine some of the elementary ideas concerning vectors. The reason for this introduction to vectors is that many concepts in science, for example, displacement, velocity, force, acceleration, have a size or magnitude, but also they have associated with them the idea of a direction. And it is obviously more convenient to represent both quantities by just one symbol. That is the vector.

Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude. This is shown in Panel 1. . If we denote one end of the arrow by the origin O and the tip of the arrow by Q. Then the vector may be represented algebraically by OQ. |

Panel 1 |

This is often simplified to just . The line and arrow above the Q are there to indicate that the symbol represents a vector. Another notation is boldface type as: Q.

Note, that since a direction is implied, . Even though their lengths are identical, their directions are exactly opposite, in fact OQ = -QO.

The magnitude of a vector is denoted by absolute value signs around the vector symbol: magnitude of Q = |Q|.

The operation of addition, subtraction and multiplication of ordinary algebra can be extended to vectors with some new definitions and a few new rules. There are two fundamental definitions.

#1 Two vectors, A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points, as shown in

Panel 2. |

Panel 2 |

#2 A vector having the same magnitude as A but in the opposite direction to A is denoted by -A , as shown in Panel 3. |

Panel 3 |

We can now define vector addition. The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of B , as illustrated in Panel 4. This is sometines referred to as the...

Here we will examine some of the elementary ideas concerning vectors. The reason for this introduction to vectors is that many concepts in science, for example, displacement, velocity, force, acceleration, have a size or magnitude, but also they have associated with them the idea of a direction. And it is obviously more convenient to represent both quantities by just one symbol. That is the vector.

Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude. This is shown in Panel 1. . If we denote one end of the arrow by the origin O and the tip of the arrow by Q. Then the vector may be represented algebraically by OQ. |

Panel 1 |

This is often simplified to just . The line and arrow above the Q are there to indicate that the symbol represents a vector. Another notation is boldface type as: Q.

Note, that since a direction is implied, . Even though their lengths are identical, their directions are exactly opposite, in fact OQ = -QO.

The magnitude of a vector is denoted by absolute value signs around the vector symbol: magnitude of Q = |Q|.

The operation of addition, subtraction and multiplication of ordinary algebra can be extended to vectors with some new definitions and a few new rules. There are two fundamental definitions.

#1 Two vectors, A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points, as shown in

Panel 2. |

Panel 2 |

#2 A vector having the same magnitude as A but in the opposite direction to A is denoted by -A , as shown in Panel 3. |

Panel 3 |

We can now define vector addition. The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of B , as illustrated in Panel 4. This is sometines referred to as the...

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