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);

vv

vuuprojv

2

• 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