Unit Vectors (Mechanics)

  1. I've been reading and studying from 'Engineering Mechanics - STATICS 5th edition' by Beford and Fowler and it says that the definition of the unit vector e is that it has a magnitude of 1.
    Then e.g. V = |V|e
    Then isn't it just V = |V|?
    I still find unit vectors pointless/confusing.
    I need some enlightment on this.
     
  2. jcsd
  3. James R

    James R 562
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    A bold-face letter represents a vector. Therefore V is a vector.

    A plain-face letter (unbolded) is a scalar quantity (which has no direction). So, V is a scalar.

    Now, if you write V = |V| then you're saying "the scalar V is equal to the magnitude of the vector V".

    But you cannot write V = V, obviously, since one object is a vector and the other is a scalar.

    So, what does this mean:

    V = |V|e ?

    What this says is "vector V equals the modulus of vector V (i.e. a scalar having the magnitude of vector V) multiplied by the unit vector e". Recall that a vector multiplied by a scalar gives a vector.

    Writing vector V this way separates out the magnitude of the vector from its direction. The magnitude is |V| = V, and the direction is the direction of e.

    Does that help?
     
  4. This and the explanation in my tutorial helped me clarify things.
    Also to put it short, do we use unit vectors to specify a certain finite length or interval within a vector of total length V?
     
  5. James R

    James R 562
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    Unit vectors are most commonly used to specify directions of vectors. Also, they are useful in setting up coordinate systems, so that arbitrary vectors can be written in component form.

    For example, a vector in two dimensions might be written as

    [tex]\mathbf{v} = 3 \hat{i} + 4 \hat{j}[/tex]

    where [itex]\hat{i}[/itex] and [itex]\hat{j}[/itex] are unit vectors in the x and y directions. Vector v in this case is a vector of length (magnitude) 5 units, and can be constructed by adding a vector of length 3 pointing in the same direction as the positive x axis and a vector of length 4 pointing in the same direction as the positive y axis.
     
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