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⟩
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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.
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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.
==== 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:
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{{ 184_notes:dotproducta.png?125|A dot product multiplies the parallel parts of two vectors using the angle between them.}}
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[{{ 184_notes:dotproducta.png?125|A dot product multiplies the parallel parts of two vectors using the angle between them.}}]
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{{ 184_notes:dotproductb.png?150|A dot product multiplies the parallel parts of two vectors using the angle between them.}}
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[{{ 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 ===
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{{ 184_notes:crossproducta.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}
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[{{ 184_notes:crossproducta.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}]
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{{ 184_notes:crossproductb.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}
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[{{ 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: