VECTORS - What is a UNIT VECTOR?

By vector definition, a UNIT VECTOR is one in which the Magnitude equals 1.  (To review how to calculate Magnitude, refer to thenormalgenius.blogspot.com article "Vectors: What is Magnitude?," published on 8/19/13.) 

In working with vectors, it is sometimes simpler to use, or the equation calls for, a unit vector in standard position, i.e., originating from (0, 0).  Transforming a given vector into a friendly unit vector is easily envisioned by relating it to common algebra and geometry.

Think of a vector as the hypotenuse of a right triangle.  Draw sides parallel to the x axis and y axis.

If you start with a right triangle (5,12,13 for example) you would need to multiply the hypotenuse by its reciprocal (the multiplicative inverse - one of the properties of real numbers) in order to get it back to 1 (the multiplicative identity).  Dividing each of the legs by 13 (the hypotenuse) would give you the coordinates that would create a special right triangle with hypotenuse of 1 (5/13, 12/13, 1).

Try it with any right triangle:

         3,4,5 transformed to a right triangle with hypotenuse of
         length 1 would be   3/5, 4/5, 1.

         x, x√3, 2x gives you sides of length (x / 2x) or 1/2
         and (x√3 / 2x) or √3 / 2.

         These numbers sound a little familiar, don’t they?  I’m thinking Trigonometry -- Sine and Cosine of 30° and 60° angles.  And they coincide with the 30°- 60° - 90° right triangle that has sides equal to our example and the hypotenuse terminating on the circumference of a UNIT circle.


So when the study of vectors comes up and you are asked to determine whether a vector is a UNIT VECTOR, if the magnitude is 1, you know it is.  For scalar or vector components, you will want to change a vector INTO a UNIT VECTOR: divide x and y by the magnitude.  In the not-so-distant future, as part of the study of vectors in Precalculus, you'll discover equations which contain terms that look like this...

and you'll recognize the unit vectors.  A discussion of components will highlight these terms in an upcoming blog article.