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- Except when velocities approach the speed of light, or in large gravitational fields, motions in any three perpendicular directions are completely independent.
- A
**vector**has one**component**for each independent direction of interest. In three dimensions,**V**= (v_{x}, v_{y}, v_{z})= v

_{x}**i**+ v_{y}**j**+ v_{z}**k**,**i**,**j**and**k**are**unit vectors**(of length 1) in the x, y and z directions, respectively. If**i**and**j**are in their usual positions in the plane of the page, the direction of**k**is*out of the page, pointing towards you:*Position, velocity and acceleration are all vectors. We will denote vectors in

**red bold type**in these notes. - The sum or difference of two vectors
**U**and**V**is**U**±**V**= (u_{x}± v_{x}, u_{y}± v_{y}, u_{z}± v_{z}).**resultant**.The vector from the point

**u**to the point**v**is**v - u**. - The
**inner product**or**dot product**of two vectors**U**and**V**is**U**⋅**V**≡ u_{x}* v_{x}+ u_{y}* v_{y}+ u_{z}* v_{z}**U**and**V**are perpendicular. Think of the dot product as the component of**U**along the direction of**V**, times the magnitude of**V**.The length (or

**magnitude**) of a vector**V**isV ≡ |V| = √ (

**V**⋅**V**).**V**/ V is the unit vector in the direction of**V**.*Note that the sum of vector magnitudes is in general meaningless.* - In two dimensions, using ordinary polar coordinates (φ measured counterclockwise from the positive x axis,
0 ≤ φ < 2π),
v

Note that tan φ = v_{x}= V cos φ, andv

_{y}= V sin φ._{y}/ v_{x}.*Note that these equations are only valid for φ measured counterclockwise from the positive x axis.*If you label a different angle as φ, you may have v_{x}= V sin φ, etc.See this applet for examples.

- When motion is on an
**inclined plane**,**resolve**the relevant vectors into components**parallel**and**normal**to the plane: - The
**vector product**or**cross product**of two vectors**U**and**V**is**U ⊗ V**≡ (U_{y}V_{z}- U_{z}V_{y}, U_{z}V_{x}- U_{x}V_{z}, U_{x}V_{y}- U_{y}V_{x})**U**and**V**. This implies**i ⊗ j**=**k****j ⊗ i**= -**k****i ⊗ i**= 0**j ⊗ k**=**i****k ⊗ j**= -**i****j ⊗ j**= 0**k ⊗ i**=**j****i ⊗ k**= -**j****k ⊗ k**= 0**U**and**V**are (anti-)parallel. Think of the cross product as the component of**U***perpendicular*to the direction of**V**, times the magnitude of**V**.

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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.