College Trigonometry, 2011a

11.8 Vectors 1021
‖k⃗v‖ = ‖k 〈v 1 ,v 2 〉‖
= ‖〈kv 1 ,kv 2 〉‖ Definition of scalar multiplication
√
= (kv 1 ) 2 +(kv 2 ) 2 Definition of magnitude
= √ k 2 v 2 1
+ k 2 v 2 2
= √ k 2 (v 2 1
+ v 2 2
)
= √ k 2√ v 2 1
+ v 2 2
Product Rule for Radicals
= |k| √ v 2 1
+ v 2 2
Since √ k 2 = |k|
= |k|‖⃗v‖
The equation ⃗v = ‖⃗v‖ˆv in Theorem 11.20 is a consequence of the definitions of ‖⃗v‖ and ˆv and was
worked out in the discussion just prior to Definition 11.8 on page 1020. In words, the equation
⃗v = ‖⃗v‖ˆv says that any given vector is the product of its magnitude and its direction – an important
(
concept to keep in mind when studying and using vectors. The equation ˆv = 1
‖⃗v‖
solving ⃗v = ‖⃗v‖ˆv for ˆv by multiplying 11 both sides of the equation by 1
of Theorem 11.19. We are overdue for an example.
Example 11.8.4.
‖⃗v‖
)
⃗v is a result of
and using the properties
1. Find the component form of the vector ⃗v with ‖⃗v‖ = 5 so that when ⃗v is plotted in standard
position, it lies in Quadrant II and makes a 60 ◦ angle 12 with the negative x-axis.
2. For ⃗v = 〈 3, −3 √ 3 〉 , find ‖⃗v‖ and θ, 0≤ θ