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| 184_notes:math_review [2018/05/17 13:36] – [Unit Vectors] curdemma | 184_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]]*/ |
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| ===== Math Review ===== | ===== Math Review ===== |
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| ==== 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 }}] |
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| {{ 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⟩
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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⟩
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| //**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.}}] |
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| - **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.}}] |
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| 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: |