College Trigonometry, 2011a

1024 Applications of Trigonometry
If ⃗v is a unit vector, ( then ) necessarily, ⃗v = ‖⃗v‖ˆv =1· ˆv =ˆv. Conversely, we leave it as an exercise 15
to show that ˆv = 1
‖⃗v‖
⃗v is a unit vector for any nonzero vector ⃗v. In practice, if ⃗v is a unit
vector we write it as ˆv as opposed to ⃗v because we have reserved the ‘ˆ’ notation for unit vectors.
1
The process of multiplying a nonzero vector by the factor
‖⃗v‖
to produce a unit vector is called
‘normalizing the vector,’ and the resulting vector ˆv is called the ‘unit vector in the direction of
⃗v’. The terminal points of unit vectors, when plotted in standard position, lie on the Unit Circle.
(You should take the time to show this.) As a result, we visualize normalizing a nonzero vector ⃗v
as shrinking 16 its terminal point, when plotted in standard position, back to the Unit Circle.
y
⃗v
1
ˆv
−1 1
x
−1
( )
Visualizing vector normalization ˆv = 1
‖⃗v‖
⃗v
Of all of the unit vectors, two deserve special mention.
Definition 11.10. The Principal Unit Vectors:
ˆ The vector î is defined by î = 〈1, 0〉
ˆ The vector ĵ is defined by î = 〈0, 1〉
We can think of the vector î as representing the positive x-direction, while ĵ represents the positive
y-direction. We have the following ‘decomposition’ theorem. 17
Theorem 11.21. Principal Vector Decomposition Theorem:
component form ⃗v = 〈v 1 ,v 2 〉.Then⃗v = v 1 î + v 2 ĵ.
Let ⃗v be a vector with
The proof of Theorem 11.21 is straightforward. Since î = 〈1, 0〉 and ĵ = 〈0, 1〉, wehavefromthe
definition of scalar multiplication and vector addition that
v 1 î + v 2 ĵ = v 1 〈1, 0〉 + v 2 〈0, 1〉 = 〈v 1 , 0〉 + 〈0,v 2 〉 = 〈v 1 ,v 2 〉 = ⃗v
(
15 One proof uses the properties of scalar multiplication and magnitude. If ⃗v ≠ ⃗0, consider ‖ˆv‖ = ∣∣
the fact that ‖⃗v‖ ≥0 is a scalar and consider factoring.
16 ...if ‖⃗v‖ > 1 ...
17 We will see a generalization of Theorem 11.21 in Section 11.9. Stay tuned!
1
‖⃗v‖
)
∣
⃗v ∣∣. Use