A good answer might be:

  1. Is it possible for the vector kv to be longer than v?
    Yes, as in the previous example.
  2. Is it possible for the vector kv to be longer than w?
    3. No, the longest kv can be is | w |, when w and v are co-linear.

Groan! Another Example

 

Notice that the projection of w onto v does NOT depend on the length of v. Only the direction of vector v affects the results. The length of the projection is | w | cos θ, where θ is the angle between the vectors. Since the cosine function is never more than one, the length of the projection will never be longer than the projected vector.

(These are not facts to be memorized; the idea is to practice visualizing what is going on by thinking about such things). And nothing better for visualization than other example.



QUESTION 12:

To make this example especially hard, I've changed the colors. Project w onto v by examining the diagram.

  w   =   (_____, _____)T       v   =   (_____, _____)T
kv   =   (_____, _____)T       u   =   (_____, _____)T