Unit Vectors from Others

The unit vector in the same direction as a given vector

Given an arbitrary vector V = Ob, suppose its tail is at the origin. The unit circle cuts V at a point a. The vector U from O to a is a unit vector in the same direction as V.

What's U? As we've seen, U = (cos t, sin t), where t is the angle which V makes with the positive x-axis.

Another approach: V/|V| is a positive multiple of V (|V| is the length of V). So V/|V| points in the same direction as V. What is the length of V/|V|?

|cV| = c|V| for positive numbers c, so |V/|V|| = |V|/|V| = 1. V/|V| is the unit vector pointing in the same direction as V.