1. Juan is sitting on a sled on the side of a hill inclined at 45 ,
with Rafaela holding Juan from sliding down the hill. The
combined weight of Juan and the sled is 140 lb. What force is
required for Rafaela to keep the sled from sliding down the hill?

2. Applications of
Vector Projection
6.2b

3.  If I pull with force u, then the effective force acting
Consider the situation of pulling a box along the ground with
a rope that is not parallel to the ground: u v
proj u v
Obviously, the box will move in the direction of v
 If I pull with force u, then the effective force acting
on the box is in the direction of v, the vector
projection of u onto v.

4. Juan is sitting on a sled on the side of a hill inclined at 45 ,
with Rafaela holding Juan from sliding down the hill. The
combined weight of Juan and the sled is 140 lb. What force is
required for Rafaela to keep the sled from sliding down the hill?
F = 70, , |F| = = lb.
 Remember this example when you do #45 and 46 on the HW!!!

5. Proctor is once again sitting on an incline, this time with a
29 angle of inclination. If Proctor weighs 199 lb, what is the
force required to keep Proctor from sliding down the hill?
What is the component of Proctor’s weight that is perp. to
the hill? (use vector projection to solve)
|F| = lb, |F | = lb

6. A 120 pound force F is pulling a box up an inclined plane. If
F makes an angle of 15 with the plane, and the plane makes
an angle of 21 with the horizontal, find the magnitude of the
effective force pulling the box up the plane. 15
120 lb 21
F = , , |F | = lb. R R

7. From physics-land… Do you think work is a scalar or a vector???
In general, work is defined as a force applied over a distance
Do you think work is a scalar or a vector???
In vector notation, if a force F is acting in the same direction
of AB, then the work (W) done by F in moving an object from
A to B is
W = |F| |AB|

8. From physics-land… W = F AB = |F| |AB| cos 0
But what if the force is not acting in
the same direction as the motion?...
If a force F is acting in any direction, then the work (W) done
by F in moving an object from A to B is
W = F AB
= |F| |AB| cos 0
where 0 is the angle between F and AB.

9. Let’s solve this one using both versions of the formula…
Practice Problems
Find the work done by a 10 pound force acting in the direction
1, 2 in moving an object 3 feet from (0, 0) to (3, 0).
Let’s solve this one using both versions of the formula…
First, a diagram…
Component form of each vector:
W = foot-pounds

10. Let’s solve this one using both versions of the formula…
Practice Problems
Find the work done by a 10 pound force acting in the direction
1, 2 in moving an object 3 feet from (0, 0) to (3, 0).
Let’s solve this one using both versions of the formula…
Magnitude of each vector:
Angle between the vectors:
W = foot-pounds

11. Practice Problems W = 74.963 N-m
Find the work done by a force F of 24 Newtons acting in the
direction 4, 5 in moving an object 5 meters from (0, 0) to
(5, 0).
W = N-m

12. Practice Problems W = 2985 ft-lb
Find the work done by lifting a 199 pound math teacher 15 feet.
W = 2985 ft-lb

13. Practice Problems W = 245.145 ft-lb
Find the work done by a force F of 50 pounds acting in the
direction 2, 3 in moving an object 5 feet from (0, 0) to a
point in the first quadrant along the line y = x.
First, the diagram…
Component form of each vector:
W = ft-lb

14. Practice Problems W = 201.944 ft-lb
The angle between a 75 pound force F and AB is 60 , where
A = (–1, 1) and B = (4, 3). Find the work done by F in moving
an object from A to B.
W = ft-lb