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It Winds From Chicago To LA,
More Than Two Thousand Miles All The Way
Vector Addition

 
 Sometimes we need to add two vectors together.
 If we are very lucky, they will be heading in the same direction.
 
 In that case, we just add them together
 and the direction stays the same.
 

 

 
 That rarely happens.
 
 The next easiest case is if the vectors are headed

 in exactly the opposite direction.

 
 In that case we just subtract the smaller one from the larger one.
 The direction of what's left is the same as the original larger one.
 

 

 
 So all of that is fine and pretty easy to deal with.
 
 But what if the two vectors are headed at crazy angles to each other?
 
 We can't easily add or subtract vectors 
 unless they are pointing in the same or opposite directions.
 So we make the crazy angle vectors point in the same or opposite directions.
 
 We do that by dividing each vector into it's component parts
 that run along the X and Y axis.
 
 Then we can add or subtract the vectors that are along the same axis.
 
 Example:
 
 Say we want to add together two vectors. 
 One goes for two miles 30 north of east (compass heading 60).
 The other one goes for two miles 45 north of east (compass heading 45).
 

 
 If we call North the positive Y direction,
 and call East the positive X direction,
 the components of the vectors are ...
 

 
 So the sums of the component vectors 
 in the X (East) and Y (North) directions are ...
 

 X = 1.732 mi + 1.414 mi = 3.146 mi

Y = 1 mi + 1.414 mi = 2.414 mi

 
 The angle of this new combined vector can be found
 by using the tangent function on these X and Y values ...
 

 
 We really did just just place these vectors head to tail and add them. 
 But doing it graphically just doesn't get you the length or angle.
 

 

   copyright 2005 Bruce Kirkpatrick

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