The applet below shows two vectors: and , where , for some scalar . You can change vector , by adjusting the sliders for magnitude and direction, and you can change vector by adjusting the slider for .
Play around with the applet for a bit, until you understand the relationships. Then, answer the questions below.

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