Calculus II For Dummies
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Every nonzero vector has a corresponding unit vector, which has the same direction as that vector but a magnitude of 1. To find the unit vector u of the vector


you divide that vector by its magnitude as follows:


Note that this formula uses scalar multiplication, because the numerator is a vector and the denominator is a scalar.

A scalar is just a fancy word for a real number. The name arises because a scalar scales a vector — that is, it changes the scale of a vector. For example, the real number 2 scales the vector v by a factor of 2 so that 2v is twice as long as v.

As you may guess from its name, the unit vector is a vector.

For example, to find the unit vector u of the vector


you first calculate its magnitude |q|:


Now use the previous formula to calculate the unit vector:


You can check that the magnitude of resulting vector u really is 1 as follows:


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Mark Zegarelli, a math tutor and writer with 25 years of professional experience, delights in making technical information crystal clear — and fun — for average readers. He is the author of Logic For Dummies and Basic Math & Pre-Algebra For Dummies.

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