The applet below shows two vectors: [math]u[/math] and [math]w[/math], where [math]w=t\cdot u[/math], for some scalar [math]t[/math]. You can change vector [math]u[/math], by adjusting the sliders for magnitude and direction, and you can change vector [math]w[/math] by adjusting the slider for [math]t[/math]. [br][br]Play around with the applet for a bit, until you understand the relationships. Then, answer the questions below.

[b]Question 1:[/b] We will first make vector [b]w[/b] a unit vector. The question is, what scalar do we multiply to [b]u[/b], to get a unit vector (in the same direction)?[br][b]a)[/b] What is a unit vector?[br][br][b]b) [/b]Set the magnitude of [b]u[/b] to 2. Then, adjust t, to make [b]w[/b] a unit vector. What t value is this?[br][br][b]c)[/b] Repeat the above process, for various magnitudes of [b]u[/b].[br][br][b]d)[/b] Hence, if the magnitude of vector [b]u[/b] is [b]|u|[/b], write an equation for [b]w[/b], to make [b]w[/b] a unit vector in the same direction as [b]u[/b].[br][br][br][b]Question 2:[/b] We will now try to generalize this idea. Say we want a vector in the same direction as [b]u[/b], but length k?[br][b]a)[/b] Set the magnitude of [b]u[/b] to be 5. Then, adjust t to make [b]w [/b]have magnitude 10 (with the same direction as [b]u[/b]. What t value is this?[br][br][b]b)[/b] Repeat the above process, for various magnitudes of [b]u[/b]. Also, change the magnitude you want [b]w[/b] to be in (instead of 10).[br][br][b]c) [/b]Hence, if the magnitude of vector [b]u[/b] is [b]|u|[/b], write an equation for [b]w[/b], to make [b]w[/b] have length k, in the same direction as [b]u[/b].[br][br][br][b]Question 3: [/b]We can also extend this, to parallel vectors.[br][b]a)[/b] What does it mean for two vectors to be parallel?[br][br][b]b)[/b] Hence, if the magnitude of vector [b]u[/b] is [b]|u|[/b], write an equation for [b]w[/b], to make [b]w[/b] have length k, [b]parallel[/b] to [b]u[/b].