We can use this to visualize scalar multiplication.[br][br]Drag around vector [math]\begin{matrix}\rightharpoonup\\u\end{matrix}[/math], and the initial point of the new vector. Then, choose the scalar [math]k[/math] you want to multiply [math]\begin{matrix}\rightharpoonup\\u\end{matrix}[/math] by, and drag the top slider to show the scalar multiple.[br][br]Then, answer the questions below, using the applet to try different cases until you convince yourself you are correct.

[b]Question 1:[/b] For real numbers, we have a number 1 such that [math]1\cdot x=x\cdot1=x[/math]. For vectors, what is [math]1\cdot u[/math]?[br][br][b]Question 2: [/b]For real numbers, we have a number 0 such that [math]0\cdot x=x\cdot0=0[/math]. For vectors, what is [math]0\cdot u?[/math]? [b]Hint: [/b]Be very specific in your answer. This is different from real numbers.[br][br]Now, considering multiple additions is okay to visualize, but it would make it difficult to imagine when [math]k[/math] is not a natural number. Instead, let's define scalar multiplication for all real numbers [math]k[/math].[br][br][b]Question 3: [/b]Create a definition for scalar multiplication.[br][b]Hint: [/b]Include the following explanations:[br] - What changes, and by how much? [br] - What stays the same?[br] - What happens if [math]k[/math] is negative? 0?[br][br][b]Question 4: [/b]Using your definition above, consider vector [math]u[/math]. Describe in terms of [math]u[/math]: [math]5u[/math] and [math]-3u[/math].