From the image, we see two vectors: $\vec{u}$ and $\vec{v}$. The question asks by what scalar multiple vector $\vec{u}$ has been scaled to produce vector $\vec{v}$.

Key observations:

1. Length change: It seems that $\vec{v}$ is approximately twice the length of $\vec{u}$, but in the opposite direction.
2. Direction: Since $\vec{v}$ points in the opposite direction to $\vec{u}$, the scalar must be negative.

Therefore, the scalar that $\vec{u}$ has been multiplied by to obtain $\vec{v}$ is approximately $-2$.

The correct answer is $-2$.

Would you like further clarification? Here are 5 questions to expand on this topic:

1. How do scalar multiplications affect the direction of vectors?
2. What is the geometric significance of a negative scalar in vector multiplication?
3. How can you calculate the magnitude of a vector after a scalar multiplication?
4. Can scalar multiplication change a vector's magnitude without affecting its direction?
5. What is the relationship between scalar products and vector dot products?

Tip: When a vector's direction changes, check if the scalar is negative.