6.4 Properties of Vectors
Properties of Vector Addition
 Commutative Property:
   
ab ba
 Associative Property:
a  b   c  a  b  c 
 Adding the Zero Vector:
  
a0a

( 0 is called the identity element for vector addition)
Properties of Scalar Multiplication
 Distributive Law:
k (u  v )  ku  kv
 Scalar Distributive Law:
 k  m u  ku  mu
 Associative Law:
k (mu )  (km)u
 Commutative Law:
ku  uk
 Identity Law:
1u  u
(The scalar 1 is called the identity element for scalar multiplication)
Basically, these “laws” should look very familiar. They are essentially the laws of Algebra.
When performing operations on vectors, you can temporarily not write the arrowheads above each vector
during the simplification stage, but you must put them in the final answer.
You will see what I mean when you study Example 1 c) d) and e) and Example 2.
Example 1 – Simplify each expression (perform the stated operations)
a)
⃗
𝑢
⃗ + 𝑢
⃗ + 𝑢
⃗ = 𝟑𝒖
b)
⃗ − 𝟐𝒗
⃗
2𝑢
⃗ − 3𝑣 − 3𝑢
⃗ + 𝑣 = −𝒖
c)
⃗
3(𝑢
⃗ + 𝑣 ) − 3(𝑢
⃗ − 𝑣 ) = 𝟑𝒖 + 𝟑𝒗 − 𝟑𝒖 + 𝟑𝒗 = 𝟔𝒗
d)
⃗ − 𝟑𝒗
⃗
3𝑢
⃗ + 2𝑣 − 2(𝑣 − 𝑢
⃗ ) + (−3𝑣 ) = 𝟑𝒖 + 𝟐𝒗 − 𝟐𝒗 + 𝟐𝒖 − 𝟑𝒗 = 𝟓𝒖
e)
⃗ + 𝟕𝒗
⃗
− (𝑢
⃗ + 𝑣 ) − 4(𝑢
⃗ − 2𝑣 ) = −𝒖 − 𝒗 − 𝟒𝒖 + 𝟖𝒗 = −𝟓𝒖
----------------------------------------------------------------------------------------------------------------------------------
Textbook Practice: pg. 307 # 6 , 7 , 8
SOLUTIONS
Example 2 – If 𝑎 = 3𝑥 + 2𝑦 and 𝑏⃗ = 5𝑥 − 4𝑦 , then:
⃗.
a) Express 𝑥 in terms of 𝑎 and 𝑏
The vector that is not mentioned is vector 𝑦. We use elimination, to get rid of vector 𝑦.
⃗.
b) Express 𝑦 in terms of 𝑎 and 𝑏
The vector that is not mentioned is vector 𝑥 . We use elimination, to get rid of vector 𝑥 .
Textbook Practice: pg. 307 # 9 , 10 , 11