Dot Product of Two Vectors
a, b  c, d
ac+bd
Example 1
4,5  2,3
Example 2
u= 1,3
(u  v ) w
v= 2,4
w= 1,2
u 2v
Angle Between Two Vectors
a b
cos 
a b
Example 3
a b
cos 
a b
Find the angle between the two vectors 3,4 and 1,5
a b
Use the formula cos 
a b
to determine the
the angle between two vectors whose dot product is
zero.
~Flashback~
A=30º
a=12
B=45º
b=
C=
c=
This is an oblique triangle with what given
information?
What do you use to solve for the sides?
What is angle C and sides b & c ?
6.4 Finding
Vector
Components &
Work
Added Two vectors
projv u  (
uv
v
2
)v
Find the projection of u=<3,5>
onto v=<6,2>.
Write u as the sum of two
orthogonal vectors, one of which
is projvu proj u  ( u  v )v
v
v
2
1500 lb. boat at a 20 degree incline.
projv u  (
uv
v
2
)v
200 lb. cart on a 30 degree incline.
projv u  (
uv
v
2
)v
W  F  DWork
A force of 45 lbs. in the direction of 30 degrees
above the horizontal is required to slide a table
across the floor. Find the work done if the table is
dragged 20 feet.
W  FD
50 lb. force,
12 ft. wide door
58) A force of 45 pounds in the direction of 30 degrees above the
horizontal is required to slide a table across a floor. Find the work done if
the table is dragged 20 feet.
59) A toy wagon is pulled by exerting a force of 20 pounds on a handle
that makes a 25 degree angle with the horizontal. Find the work done in
pulling the wagon 20 feet.
60) A mover exerts a horizontal force of 25 pounds on a crate as it is
pushed up a ramp that is 12 feet long and inclined at an angle of 20
degrees above the horizontal. Find the work done on the crate.