Component Vectors

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The Unit Vector

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1 of 3 videos by James S.

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.

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  • 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)
    enter image source here
    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?

    enter image source here

    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?

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