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Unit vectors and direction cosines

  1. Nov 14, 2012 #1
    Let [itex]\vec{A}[/itex] represent any nonzero vector. Why is [itex]\frac{\vec{A}}{A}[/itex] a unit vector and what is its direction? If θ is the angle that [itex]\vec{A}[/itex] makes with the positive x-axis, explain why [itex]\frac{\vec{A}}{A}\cdot\hat{i}[/itex] is called the direction cosine for that axis.


    I am self-studying and this question has me stumped. I am familiar with the formula for a unit vector but I don't know why it's true and I have never really heard of a direction cosine. Could anyone give me some hints, perhaps?
     
  2. jcsd
  3. Nov 14, 2012 #2
    Okay, I've thought about the first part a little more. A unit vector is a vector whose magnitude is 1. So we must show that the magnitude of [itex]\frac{\vec{A}}{A}[/itex] is 1. Through algebraic manipulation I can show that pretty easily. I still don't know about the direction or the direction cosines, though.
     
  4. Nov 14, 2012 #3

    haruspex

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    It isn't made clear, but presumably [itex]A = |\vec{A}|[/itex]. Since that is a scalar, what does dividing a vector by it do to the vector's direction?
    If [itex]\hat{i}[/itex] is a unit vector at angle θ to [itex]\vec{A}[/itex], what is the magnitude of [itex]\vec{A}\cdot\hat{i}[/itex]?
     
  5. Nov 15, 2012 #4
    Ah, okay. (Yes, [itex]A[/itex] means [itex]|\vec{A}|[/itex].) Since it's a scalar, then the direction will remain the same.

    [itex]\vec{A}\cdot \hat{i}[/itex] is equivalent to [itex]|\vec{A}|\cdot|\hat{i}|\cdot\cos\theta[/itex], or, since [itex]\hat{i}[/itex] is has a magnitude of 1, just [itex]|\vec{A}|\cos\theta[/itex]. I don't understand what is meant by the magnitude of this dot product, though. It's just a scalar, right?

    Thanks so much for your help!!
     
  6. Nov 15, 2012 #5
    I think I've got it. The magnitude of [itex]\frac{\vec{A}}{|\vec{A}|}[/itex] is just 1, and so is the magnitude of [itex]\hat{i}[/itex]. So [itex]\frac{\vec{A}}{|\vec{A}|}\cdot\hat{i}[/itex] is just [itex]\cos\theta[/itex].

    So all that is just a long way of saying that the direction cosine is just the cosine of the angle between the x-axis and the vector?
     
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