184_notes:math_review

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184_notes:math_review [2018/05/17 13:31] – [Vector Notation] curdemma184_notes:math_review [2018/05/17 13:56] – [Vector Addition] curdemma
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 $$\vec{a} = |\vec{a}|\hat{a}$$ $$\vec{a} = |\vec{a}|\hat{a}$$
  
-We also use unit vectors to describe the x, y, and z coordinate directions. This are represented by an $\hat{x}$, $\hat{y}$, and $\hat{z}$ or by an $\hat{i}$, $\hat{j}$, and $\hat{k}$. 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 $\hat{x}$, $\hat{y}$, and $\hat{z}$ or by an $\hat{i}$, $\hat{j}$, and $\hat{k}$. 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).
 $$\vec{a} = a_x\hat{x}+a_y\hat{y}+a_z\hat{z}$$ $$\vec{a} = a_x\hat{x}+a_y\hat{y}+a_z\hat{z}$$
 $$\vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$$ $$\vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$$
  
 ==== 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:
 $$ \vec{a} +\vec{b} = \langle a_x, a_y, a_z \rangle + \langle b_x, b_y, b_z \rangle = \langle a_x+b_x, a_y+b_y, a_z+b_z \rangle  $$ $$ \vec{a} +\vec{b} = \langle a_x, a_y, a_z \rangle + \langle b_x, b_y, b_z \rangle = \langle a_x+b_x, a_y+b_y, a_z+b_z \rangle  $$
 $$ \vec{a} - \vec{b} = \langle a_x, a_y, a_z \rangle - \langle b_x, b_y, b_z \rangle = \langle a_x-b_x, a_y-b_y, a_z-b_z \rangle $$  $$ \vec{a} - \vec{b} = \langle a_x, a_y, a_z \rangle - \langle b_x, b_y, b_z \rangle = \langle a_x-b_x, a_y-b_y, a_z-b_z \rangle $$ 
-//**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 $\vec{a}=\langle a_x, a_y, a_z \rangle$ and $\vec{b}=\langle b_x, b_y, b_z\rangle$, 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 $\vec{a}=\langle a_x, a_y, a_z \rangle$ and $\vec{b}=\langle b_x, b_y, b_z\rangle$, 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:
  • 184_notes/math_review.txt
  • Last modified: 2020/08/24 19:30
  • by dmcpadden