Vector Notation in Nolting Theoretical Physics 1

In summary, on pages 60-61 of Nolting Theoretical Physics 1, there is a notation for vectors multiplied by a scalar where two little up arrows are used if the scalar is greater than zero and a little up arrow followed by a little down arrow is used if the scalar is less than zero. This notation means that the resulting vector is parallel to the original vector if the scalar is positive and anti-parallel if the scalar is negative. This notation may seem unfamiliar, but it is important to understand in order to fully grasp the concept of vectors.
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Teclis
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On pg. 60 of Nolting Theoretical Physics 1 for the definition of a vector multiplied by a scalar the book shows two little up arrows if the scalar is greater than zero and an little up arrow and then a little down arrow if the scalar is less than zero. Then again on pg. 61 for definition 1.139 there are two little up arrows (similar to particle spin notation) in between the unit vector and the vector.

What do these little pairs of arrows mean? I have never seen this notation for vectors before. Could someone please explain their semantics?
 
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For reference, this is the excerpt of that definition:
upload_2018-3-13_8-42-7.png


This property is saying that ##\alpha\bf a## is parallel to (i.e., same direction as) ##\bf a## if ##\alpha > 0## and anti-parallel to (i.e., opposite direction) ##\bf a## if ##\alpha < 0##. Not mentioning this (I did not find it anywhere) seems like a serious omission by the author. In particular as the preliminaries seem to aim on defining what the student needs to know about vectors.
 

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Related to Vector Notation in Nolting Theoretical Physics 1

1. What is vector notation?

Vector notation is a mathematical representation used to describe physical quantities that have both magnitude and direction. It is often represented by an arrow or bold letter, and can be used to represent quantities such as velocity, force, and acceleration.

2. How is vector notation used in Nolting Theoretical Physics 1?

Vector notation is used extensively in Nolting Theoretical Physics 1 to describe and solve problems in classical mechanics. It allows for a more concise and clear representation of physical quantities and their relationships, making it an essential tool for understanding fundamental concepts in physics.

3. What are the basic rules of vector notation?

The basic rules of vector notation include addition, subtraction, and multiplication by a scalar. Vectors can be added or subtracted by adding or subtracting their respective components, and can be multiplied by a scalar (a number) to change their magnitude without changing their direction.

4. What is the difference between dot and cross products in vector notation?

The dot product is a scalar quantity that results from multiplying two vectors and taking the cosine of the angle between them. It represents the magnitude of the projection of one vector onto the other. The cross product, on the other hand, is a vector quantity that results from multiplying two vectors and taking the sine of the angle between them. It represents the magnitude and direction of a vector perpendicular to both input vectors.

5. How does vector notation relate to coordinate systems?

Coordinate systems, such as Cartesian coordinates, are often used to represent vectors in a graphical form. In these systems, vectors are represented by their components in the x, y, and z directions. This allows for a visual representation of vectors and their relationships, which can aid in problem solving and understanding of physical concepts.

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