Part 3: Unit Vectors and Projections

A vector with a length of 1 is known as a unit vector. It follows that any nonzero vector v can be written uniquely in the form

v = || v||  u

(2)

where u is a unit vector. Solving for u in (2) reveals that if v ¹ 0, then

u =  v ||v||

(3)

We say that || v|| is the magnitude of v. Thus, the unit vector u is understood to represent the direction of v.       

EXAMPLE 6    Find the direction and decomposition (2) of v = á 1,Ö3 ñ .       

Solution: The length of v is || v|| = 2. Thus, the direction of v is

u =   v || v||  =  á1, 3 ñ 2  =  1 2 , 3 2

It follows that v can be written in terms of its magnitude and direction as

v = || v||  u = 2   1 2 , 3 2

       

Finally, the projection of a vector v onto a nonzero vector p is defined

projp( v) =  v·p p·p   p

(6)

Geometrically, the projection of v onto p is parallel to p.   Moreover, notice that

( v-projp( v) ) ·p   =   v -  v·p p·p p ·p   =   v·p -    v·p p·p   p·p   =   v·p-v·p   =   0

That is, v - proj_p( v) is orthogonal to p.  Geometrically, this means that proj_p(v) and v - proj_p(v) form the sides of a right triangle.

EXAMPLE 7    Find the projection of v = á 2, 0, 5 ñ onto p = á 4, 4, 2 ñ . Also, show that

w = v-projp( v)

is orthogonal to p.        

Solution: To do so, we use the formula (4) to obtain

projp( v) =  v·p p·p   p =  2·4+0·4+5·2 4·4+4·4+2·2   á 4,4,2 ñ =  18 36 á4,4,2 ñ

Thus, proj_p( v) = á2,2,1 ñ and w = á 2,0,5 ñ - á 2,2,1 ñ = á 0,-2,4 ñ . Also, proj_p(v) · w = 2(0)+2(-2) +1(4) = 0.

Projections are important in many different applications, as we will see in the exercises, in the worksheet, and in several sections of the text.

Check your reading: What || P || if P = á2,2,1 ñ ? How is this related to example 7?