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184_notes:math_review [2018/05/17 13:34] – [Vector Notation] curdemma184_notes:math_review [2020/08/24 19:30] (current) – dmcpadden
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-[[184_notes:defining_a_system|Next Page: Defining a System]]+/*[[184_notes:defining_a_system|Next Page: Defining a System]]*/
  
 ===== Math Review ===== ===== Math Review =====
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 where ax, ay, and az are the vector components in the x, y, and z direction respectively. They tell you "how much" of the vector →a is aligned with each coordinate direction. The vector itself is denoted either in bold face (typical in textbooks) or with an arrow above it.  where ax, ay, and az are the vector components in the x, y, and z direction respectively. They tell you "how much" of the vector →a is aligned with each coordinate direction. The vector itself is denoted either in bold face (typical in textbooks) or with an arrow above it. 
  
-The magnitude (or length of a vector) is a scalar quantity and is denoted by vertical lines on either side of the vector (e.g. |$$a|). It can be found by using the [[https://en.wikipedia.org/wiki/Pythagorean_theorem|Pythagorean theorem]] in three dimensions:+The magnitude (or length of a vector) is a scalar quantity and is denoted by vertical lines on either side of the vector. It can be found by using the [[https://en.wikipedia.org/wiki/Pythagorean_theorem|Pythagorean theorem]] in three dimensions:
 a=|→a|=√a2x+a2y+a2z
 a=|→a|=√a2x+a2y+a2z
  
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 →a=|→a|ˆa
 →a=|→a|ˆa
  
-We also use unit vectors to describe the x, y, and z coordinate directions. This are represented by an ˆx, ˆy, and ˆz or by an ˆi, ˆj, and ˆk. Using these coordinate unit vector, you can write any vector in terms of its components. These are common alternative ways to write vectors (as opposed to the bracket notation).+We also use unit vectors to describe the x, y, and z coordinate directions. These are represented by an ˆx, ˆy, and ˆz or by an ˆi, ˆj, and ˆk. Using these coordinate unit vectors, you can write any vector in terms of its components. These are common alternative ways to write vectors (as opposed to the bracket notation).
 →a=axˆx+ayˆy+azˆz
 →a=axˆx+ayˆy+azˆz
 →a=axˆi+ayˆj+azˆk
 →a=axˆi+ayˆj+azˆk
  
 ==== Vector Addition ==== ==== Vector Addition ====
 +[{{  course_planning:course_notes:2d_vector_addition.png?225|graphical vector addition  }}] 
 +[{{ course_planning:course_notes:2d_vector_subtraction.png?225|graphical vector subtraction }}]
 +
  
-{{ course_planning:course_notes:2d_vector_addition.png?250|graphical vector addition}} 
-{{ course_planning:course_notes:2d_vector_subtraction.png?225|graphical vector subtraction}} 
 Two vectors are added (or subtracted) component by component: Two vectors are added (or subtracted) component by component:
 →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⟩
 →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⟩
 
-//**Note: You CANNOT simply add or subtract the magnitudes.**// This disregards the direction that the vectors point in. Alternatively, you can use the [[183_notes:scalars_and_vectors#adding_&_subtracting_vectors|"tip-to-tail" method]] to add or subtract vectors if you have them drawn out graphically. +//**Note: You CANNOT simply add or subtract the magnitudes.**// This disregards the direction that the vectors point in. Alternatively, you can use the [[183_notes:scalars_and_vectors#adding_&_subtracting_vectors|"tip-to-tail" method]] to add or subtract vectors if you have them drawn out graphically. 
 ==== Vector Multiplication ==== ==== Vector Multiplication ====
  
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 There are a couple of ways to calculate the dot product: There are a couple of ways to calculate the dot product:
-{{ 184_notes:dotproducta.png?125|A dot product multiplies the parallel parts of two vectors using the angle between them.}} +[{{ 184_notes:dotproducta.png?125|A dot product multiplies the parallel parts of two vectors using the angle between them.}}] 
-{{ 184_notes:dotproductb.png?150|A dot product multiplies the parallel parts of two vectors using the angle between them.}}+[{{ 184_notes:dotproductb.png?150|A dot product multiplies the parallel parts of two vectors using the angle between them.}}]
  
 - **Using vector components** - If you have two vectors given by →a=⟨ax,ay,az⟩ and →b=⟨bx,by,bz⟩, then you can calculate the dot product by multiplying each component together and adding them together: - **Using vector components** - If you have two vectors given by →a=⟨ax,ay,az⟩ and →b=⟨bx,by,bz⟩, then you can calculate the dot product by multiplying each component together and adding them together:
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 === Cross Product === === Cross Product ===
-{{ 184_notes:crossproducta.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}} +[{{ 184_notes:crossproducta.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}] 
-{{ 184_notes:crossproductb.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}+[{{ 184_notes:crossproductb.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}]
  
 The cross product is another way to "multiply" two vectors together, which again has some important features: The cross product is another way to "multiply" two vectors together, which again has some important features:
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