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.
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...