Component Vectors

Key Questions

  • Answer:

    #" "#
    Please read the explanation.


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    How do we use the components of two vectors to find the resultant vector by adding the two vectors ?

    A Vector is defined as a quantity with both magnitude and direction.

    Two vectors are shown below:

    #color(red)(vec(OA) and vec(OB)#

    We will also be using these vectors in our example later.

    enter image source here

    #vec(OA) = hat(u)=(2 hat i+5 hat j)#

    In component form


    #vec(OB) = hat(v)=(4 hat i-8 hat j)#

    In component form


    Let us see how we can add these two vectors:

    #hat (u) + hat (v) = (2 hat i+5 hat j)+(4 hat i-8 hat j)#

    Using component form:

    #hat (u) + hat (v) = <2 ,5 >+<4-8 >#

    Add #color(red)(i# components and #color(red)(j# components together:

    #hat (u) + hat (v) = <2+4>+<5-8 >#

    #color(red)(hat (u) + hat (v) =<6, -3>#

    We can represent this solution graphically as follows:

    enter image source here

    The solution is represented by

    #color(red)(w=hat (u) + hat (v) =<6, -3>#


    #color(red)(w=hat (u) + hat (v) =(6i -3j)#

    Note: Alternative graphical solution process:

    #vec(OA)# can also be translated to the line in green (BC).


    #vec(OB)# can be translated to the line in blue (AC).

    We can see that #color(red)(w# is the solution.

    Hope it helps.

  • 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?

  • 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.