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Unit Vectors - Unit Vectors What is probably the most...

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Unit Vectors What is probably the most common mistake involving unit vectors is simply leaving their hats off. While leaving the hat off a unit vector is a nasty communication error in its own right, it also leads one to worse mistakes such as treating vectors as if they were scalars. The mathematicians have come up with a special kind of vector called a unit vector which comes in very handy in physics. By definition a unit vector has magnitude 1, with no units. By convention, a unit vector is represented by a letter marked with a circumflex. The circumflex is an accent mark that appears above the letter. It looks like an inverted “v” and is typically referred to as a “hat”. So for instance r ˆ ( read “ -hat”) is a unit vector. Let’s suppose, just to make this discussion more concrete, that r ˆ is at 36.0 ° (counterclockwise from the +x direction, in the x-y plane). Now the fact that a unit vector has a magnitude 1, with no units, means that if you multiply a unit vector by a scalar, the resulting vector has a magnitude equal to the value- with-units of the scalar. So for instance, if you multiply the vector r ˆ by 5.00 m/s, you get a velocity vector r ˆ s m 00 . 5 which has a magnitude of 5.00 m/s and points in the same direction as the unit vector r ˆ . Thus, in the case at hand, r ˆ s m 00 . 5 , means 5.00 m/s at 36.0 ° . There is a special set of three unit vectors that are exceptionally useful for problems involving vectors, namely the Cartesian coordinate axis unit vectors. There is one of them for each positive coordinate axis direction. These unit vectors are so prevalent that we give them special names.
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