Trig 4.2 Unit Vectors and Projections

Mathematical Investigations IV
Getting To the Point
Unit Vectors
Given a vector v we know how to find its magnitude |v|. We know how to find the angle that v
makes when measured from the positive x-axis, or when measured as a bearing from reference
direction north in a navigational context. Summarize these here for a vector v = <a, b>. Give
formulas in terms of a and b, assuming the vector is in the first quadrant as shown.
|v| =
(a, b)
angle with respect to the positive x-axis =
bearing from due north =
There is another way to describe the direction of a v called the unit vector in the
v-direction. The unit vector has a magnitude of 1 (hence the name “unit” vector) and points
in the same direction as v.
You may remember that we have defined
iˆ = unit vector in the positive x direction = <1, 0>
ĵ = unit vector in the positive y direction = <0, 1>
(watch the symbols carefully!)
The unit vector in the direction of v is often
denoted by the symbol v̂ and is defined as
vˆ 
Note that this implies that v | v | vˆ for any vector v .
Vectors 4.1
Rev. F08
Mathematical Investigations IV
Every non-zero vector has a unit vector associated with it. Find the unit vectors associated
with the following vectors:
<3, –4>
3 4
5 5
 3
 
 1 
5iˆ  12 ĵ
cos iˆ  sin  ĵ
<x, y>
Vectors 4.2
Rev. F08