183_notes:scalars_and_vectors

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183_notes:scalars_and_vectors [2021/01/28 20:57] – [Determining Vector Components in Two Dimensions] stumptyl183_notes:scalars_and_vectors [2024/01/30 13:50] (current) – [Vector Simulation] hallstein
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 ===== Scalars and Vectors ===== ===== Scalars and Vectors =====
  
-We often use mathematics to describe physical situations. Two types of quantities that are particularly important for describing physical systems are scalars and vectors. In the notes below, you will read about those quantities (in general) and their properties.+We often use mathematics to describe physical situations. Two types of quantities that are particularly important for describing physical systems are scalars and vectors. **In the notes below, you will read about those quantities (in general) and their properties.**
 ==== Lecture Video ==== ==== Lecture Video ====
  
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 Vectors are often represented with arrows. The end with the triangle is the "tip" or "head." The other end is called the "tail." The tail of a vector can be located anywhere; it is the difference between the tip and the tail that defines the vector itself. To the right is an example of a typical representation (a diagram) of a vector with the tip and tail labeled. We have no such diagrammatic representations for scalars. Vectors are often represented with arrows. The end with the triangle is the "tip" or "head." The other end is called the "tail." The tail of a vector can be located anywhere; it is the difference between the tip and the tail that defines the vector itself. To the right is an example of a typical representation (a diagram) of a vector with the tip and tail labeled. We have no such diagrammatic representations for scalars.
-==== Defining Vectors Mathematically ====+===== Defining Vectors Mathematically =====
  
 {{ course_planning:course_notes:scalars_vectors_new.png?300| A position vector defined in 3D space}} {{ course_planning:course_notes:scalars_vectors_new.png?300| A position vector defined in 3D space}}
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 \\ \\
-=== Length of a vector ===+==== Length of a vector ====
  
 The **magnitude** (or length) of a vector is a scalar quantity. Mathematically, we represent the magnitude of a vector like this: The **magnitude** (or length) of a vector is a scalar quantity. Mathematically, we represent the magnitude of a vector like this:
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 \\ \\
-=== Unit vector ===+==== Unit vector ====
  
 Any vector can be multiplied or divided by a scalar quantity. Often it is useful to divide a vector by its own magnitude. The result is the "unit vector." **The unit vector** is a vector with length 1, but that points in the direction of the original vector. //The unit vector has no units// (e.g., the unit vector of a position vector with units of meters has no units itself). Mathematically, we represent the unit vector like this: Any vector can be multiplied or divided by a scalar quantity. Often it is useful to divide a vector by its own magnitude. The result is the "unit vector." **The unit vector** is a vector with length 1, but that points in the direction of the original vector. //The unit vector has no units// (e.g., the unit vector of a position vector with units of meters has no units itself). Mathematically, we represent the unit vector like this:
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 //The above equations only work when the vectors are decomposed with along the x and y axis as defined in the figure to the right.// Oftentimes, an angle that is given or derived cannot make use of the simple decomposition formulae above. The geometric properties of the problem will dictate which trigonometric functions are used. //The above equations only work when the vectors are decomposed with along the x and y axis as defined in the figure to the right.// Oftentimes, an angle that is given or derived cannot make use of the simple decomposition formulae above. The geometric properties of the problem will dictate which trigonometric functions are used.
 +
 +{{youtube>WwxevEMyxFk?large}}
 ==== Adding & Subtracting Vectors ===== ==== Adding & Subtracting Vectors =====
  
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 →a−→b=⟨ax,ay,az⟩−⟨bx,by,bz⟩=⟨ax−bx,ay−by,az−bz⟩ →a−→b=⟨ax,ay,az⟩−⟨bx,by,bz⟩=⟨ax−bx,ay−by,az−bz⟩
  
-{{ course_planning:course_notes:2d_vector_addition.png?250|graphical vector addition}}+{{ 183_notes:vector_addition_new.png?250|graphical vector addition}}
 Graphically, vector addition and subtraction use the "tip-to-tail" method.  Graphically, vector addition and subtraction use the "tip-to-tail" method. 
  
 **Vector addition** For addition, place the tail of the second vector at the tip of the first vector. The vector that points from the tail of the first to the tip of the second is the sum or the "resultant" vector. The image to the right demonstrates this for two vectors, →a and →b. **Vector addition** For addition, place the tail of the second vector at the tip of the first vector. The vector that points from the tail of the first to the tip of the second is the sum or the "resultant" vector. The image to the right demonstrates this for two vectors, →a and →b.
  
-{{ course_planning:course_notes:2d_vector_subtraction.png?225|graphical vector subtraction}}+{{ 183_notes:vector_subtraction_new.png?225|graphical vector subtraction}}
 **Vector subtraction** For subtraction, draw the vector that points directly opposite of the second vector. Place the tail of this reversed second vector at the tip of the first vector. The vector that points from the tail of the first to the tip of the reversed second is the difference vector. The image to the right demonstrates this for two vectors, →a and →b. **Vector subtraction** For subtraction, draw the vector that points directly opposite of the second vector. Place the tail of this reversed second vector at the tip of the first vector. The vector that points from the tail of the first to the tip of the reversed second is the difference vector. The image to the right demonstrates this for two vectors, →a and →b.
  
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 Here's simulation that let's you play with vectors in 2D.((Credit the [[http://phet.colorado.edu|PhET Team]] at the University of Colorado for the simulation.)) If the embedded simulation doesn't work, you can find it [[http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html|on the PhET website]]. Here's simulation that let's you play with vectors in 2D.((Credit the [[http://phet.colorado.edu|PhET Team]] at the University of Colorado for the simulation.)) If the embedded simulation doesn't work, you can find it [[http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html|on the PhET website]].
  
-{{url>http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html 800px,600px | PhET Vector Simulation}}+{{url>https://phet.colorado.edu/sims/html/vector-addition/latest/vector-addition_all.html 800px,600px | PhET Vector Simulation}}
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