The DOT  PRODUCT
(Geometric Vectors)
A.
The DOT PRODUCT DEFINED
The dot product (or scalar product) of two vectors is defined as:
Note:
if 0o   < 90o, 𝑢 ∙ 𝑣 is > 0
if 90o <   180o, 𝑢 ∙ 𝑣 is < 0
if  = 90o, 𝑢 ∙ 𝑣 is 0
𝒖 ∙ 𝒗 = 𝒖 𝒗 𝒄𝒐𝒔𝜽
where 0o    180o
Ex.
Determine 𝑢 ∙ 𝑣 if:
a)
B.
𝑢 = 3, 𝑣 = 5, 𝜃 = 40𝑜
b)
𝑢 = 2, 𝑣 = 4, 𝜃 = 110𝑜
PROPERTIES of the DOT PRODUCT
1.
2.
3.
Commutative Property:
Distributive Property:
Associative Property:
(with a scalar)
𝑢∙𝑣=𝑣∙𝑢
𝑢∙ 𝑣+𝑤 =𝑢∙𝑣+𝑢∙𝑤
𝑘𝑢 ∙ 𝑣 = 𝑢 ∙ 𝑘𝑣 = 𝑘(𝑢 ∙ 𝑣)
4.
What is the dot product of two perpendicular vectors?
𝑢∙𝑣=
5.
What is the dot product of a vector with itself?
𝑢∙𝑢=
C.
DOT PRODUCT EXAMPLES
Ex 
Simplify:
2𝑎 + 3𝑏 ∙ (𝑎 − 𝑏)
Ex 
Calculate the angle between 𝑢 and 𝑣 if 𝑢 = 2, 𝑣 = 3, and 𝑢 ∙ 𝑣 = 3 3.
Ex 
If 𝑎 + 2𝑏 and 𝑎 − 3𝑏 are  and 𝑎 = 𝑏 = 1, determine 𝑎 ∙ 𝑏.
Ex 
Prove:
𝑎+𝑏
2
= 𝑎
2
+ 2𝑎 ∙ 𝑏 + 𝑏
2
D.
APPLICATIONS of the DOT PRODUCT (WORK)
work:
whenever a force acting on an object causes a displacement of the
object from one position to another, work is done
𝐹
work = 𝐹 𝑐𝑜𝑠𝜃 x 𝑠
= 𝐹 𝑠 𝑐𝑜𝑠𝜃
=𝐹∙𝑠

𝑠
horizontal component
𝐹 𝑐𝑜𝑠𝜃
W=𝑭∙𝒔
= 𝑭 𝒔 𝒄𝒐𝒔𝜽
where 𝐹 is the force acting on the object
𝑠 is the displacement caused by the force
 is the angle between 𝐹 and s
Work is a scalar quantity.
The unit of work is a joule (J).
Can work by a (–) value?
Ex 
A wagon is pulled 100m by a 30N force acting at 25o to the ground.
Calculate the work done.
Homework: p.377–378 #1, 2, 5–16 p.415 #3