We can use this to visualize scalar multiplication.
Drag around vector , and the initial point of the new vector. Then, choose the scalar you want to multiply by, and drag the top slider to show the scalar multiple.
Then, answer the questions below, using the applet to try different cases until you convince yourself you are correct.

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