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When to use unit vectors?

  1. Nov 14, 2015 #1
    I'm confused about what situations you should use unit vectors in... and it seems that when I approach the same problem using unit vectors vs. without unit vectors, I get different answers. Why?
    To illustrate my confusion, here's an example that I tried solving using unit vectors, and without unit vectors.

    For example:
    1. The problem statement, all variables and given/known data
    A particle moves in the xy plane from the origin at t=0 with an initial velocity having an x-component of 20m/s, and a y-component of -15m/s. The particle has an acceleration in the x-direction of ax=4m/s2

    What is the velocity of the particle at 5 seconds?

    2. Relevant equations

    Velocity as a function of time: vf=vi+at

    3. The attempt at a solution

    Expressed as unit vectors:
    vf = (vix+axt)i + (viy+ayt)j
    vf = (20+4t)i - 15j
    vf = [20 +4(5)]i - 15j = (40i - 15j)m/s
    Expressed into x and y components: vfx=40m/s, vfy=-15m/s
    vf = sqrt(402 + (-15)2) = 37m/s

    Expressed as "regular" vectors:
    x direction: vfx = vix+axt = 20+4t
    y direction: vfy = viy+ayt = -15 +4(0) = -15
    vf = sqrt[(20+4t)2 + (-15)2] = sqrt(625 + 160t + 16t2) = 43m/s

    EDIT: I rechecked my calculations, and realized that my final answer for the unit vector approach is also 43m/s... D'oh!
    Given my mistake, my question is now: Will a unit vector approach always give me the same answer as when I don't use unit vectors?
     
    Last edited: Nov 14, 2015
  2. jcsd
  3. Nov 14, 2015 #2

    SteamKing

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    In the last line of the 'unit vectors' calculation, check your arithmetic when you calculate the velocity at t = 5 sec.

    In the last line of your 'regular' calculation, why did you expand (20 + 4t)2 ? Why didn't you just evaluate (20 + 4t) and then square the result?
     
  4. Nov 14, 2015 #3
    Such a trivial mistake on my part! Yes, I redid my calculations and got 43m/s for both approaches. Now, will a "unit vector" approach always give me the same answer as when I don't use unit vectors?
     
  5. Nov 14, 2015 #4

    SteamKing

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    It should. You still have to keep your arithmetic checked, though.
     
  6. Nov 14, 2015 #5
    Of course. It's just a different way of writing things out. Note that you can write a vector equation with unit vectors, something you cannot do without them.

    For example, ##\vec{J}=25 \hat i+7 \hat j##
     
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