8.6.3 – Projections of Vectors

1. 8.6.3 – Projections of Vectors

2. 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”

3. Projection • To “project” a vector u onto v (basically, rotate a vector and place it on a second vector);

4. After the projection, we will also attempt to write the vector as a sum of two orthogonal vectors • How?

5. Example. Find the projection of u = {2,4} onto v = {7, -1}.

6. Example. Using the previous information, write u as a sum of two orthogonal vectors, one of which is the projection.

7. 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}

8. 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}

9. 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

10. 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?

11. 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?

12. 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?

13. Assignment • Pg. 679 • 43-48 , 53-57 odd