8.6.3 – Projections of Vectors
• In some cases, we will have to decompose a
vector into a sum of two separate vectors
– Recall; most vectors may be written as some
variation of the special unit vectors {1,0} and {0,1}
• With vectors, sometimes they may not be
pointing or oriented in the proper direction
• We can fix that by performing what is known
as a “projection” or “orthogonal projection”
Projection
• To “project” a vector u onto v (basically, rotate
a vector and place it on a second vector);


 u v 
projv u   2 v
 v 


• After the projection, we will also attempt to
write the vector as a sum of two orthogonal
vectors
• How?
• Example. Find the projection of u = {2,4} onto
v = {7, -1}.
• Example. Using the previous information,
write u as a sum of two orthogonal vectors,
one of which is the projection.
• Example. Find the projection of u onto v and
then write u as a sum of two orthogonal
vectors.
• u = {0,3}; v = {2, 6}
• Example. Find the projection of u onto v and
then write u as a sum of two orthogonal
vectors.
• u = {2, 3}; v = {-1, 5}
Word Problems
• The usefulness of vector projection comes in
handy with some types of word problems
• 1) Application of force (required force to
pull/push and object)
• 2) Work = application of force through a
particular distance
–W=F.D
• Example. A boat and trailer, which together
weight 500 pounds, are to be pulled up a
ramp that has an incline of 30 degrees. What
force is required by a vehicle to prevent the
boat and trailer from rolling down the ramp?
• Example. Ben is at the top of a hill on a sled
angled at 45 degrees, held by Madi. The
combined weight of Ben and the sled is 155
pounds. What force is required to prevent Ben
from sliding to a terrible death?
• Example. A child pulls a wagon along a
sidewalk, exerting a force of 15 pounds on the
handle. That is handle is 40 degrees from the
horizontal. If the child pulls the wagon 50 feet,
what work has been done?
• Assignment
• Pg. 679
• 43-48 , 53-57 odd