Component Vectors
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Key Questions

A vector has both magnitude (which is its length) and direction (which is its angle).
Any two dimentional vector at an angle will have a horizontal and a vertical component .
A vector written as ( 12 , 8 ) will have 12 as its horizontal component, and 8 as its vertical component, and because both components are positive, the vector is pointing to the northeastern direction. 
When two forces are combined, one with magnitude
#r_1# and direction#z_1# and the other with magnitude#r_2# and direction#z_2# , the resultant can be decomposed into horizontal and vertical components <#x,y# >, where:#x=r_1cos(z_1)+r_2cos(z_2)#
#y=r_1sin(z_1)+r_2sin(z_2)# example: a force of 20 mph at 30 degrees is combined with a force of 50 mph at 40 degrees. The resultant of the combined forces is:
<
#x, y# >#=# <#20cos(30)+50cos(40), 20sin(30)+50sin(40)# >#=# <#55.6, 45# > 
To find the magnitude of a vector using its components you use Pitagora´s Theorem.
Consider in 2 dimensions a vector
#vecv# given as:
#vecv = 5veci + 3vecj# (where#veci# and#vecj# are the unit vectors on the x and y axes)
The magnitude of this vector (or its length in geometrical sense) is given using Pitagora's Theorem, as:
#vecv =sqrt(5^2+3^2)= 5,8# The same thing applies in 3 dimensions, the only thing is to include the third component.
So if the vector is now given as:
#vecv = 5veci+ 3vecj + 2veck#
The magnitude will be:
#vecv= sqrt(5^2+3^2+2^2) = 6,2# 
Often when two processes interact we only know the component vector values and need to be able to combine these to get a desired result.
This might be more easily understood by an example:
Suppose I am trying to fly from point A to point B which is due North of point A. My plane flies at an air speed of 100 miles/hour but there is a wind blowing due West at 30 miles/hour. How many degrees East of North do I need to orient my plane to fly in a straight line to B?From the above diagram, I need to head my plane (approximately)
#17.5^o# East of North.This problem could be extended to ask:
If it is 200 miles from A to B and my plane has enough gas to fly 250 miles will I be able to make this trip?