We are asked to determine the sum of the given expression, which involves scalar multiplication and vector addition. The expression is:

$0 \cdot \begin{pmatrix} -5 \\ 5 \\ 0 \end{pmatrix} + (-4) \cdot \begin{pmatrix} -2 \\ -2 \\ 3 \end{pmatrix}$

Step 1: Multiply the scalars by the vectors.

1. Multiply the first vector by $0$:

$0 \cdot \begin{pmatrix} -5 \\ 5 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$

2. Multiply the second vector by $-4$:

$-4 \cdot \begin{pmatrix} -2 \\ -2 \\ 3 \end{pmatrix} = \begin{pmatrix} (-4) \cdot (-2) \\ (-4) \cdot (-2) \\ (-4) \cdot 3 \end{pmatrix} = \begin{pmatrix} 8 \\ 8 \\ -12 \end{pmatrix}$

Step 2: Add the resulting vectors.

$\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 8 \\ 8 \\ -12 \end{pmatrix} = \begin{pmatrix} 8 \\ 8 \\ -12 \end{pmatrix}$

Final Answer:

$\begin{pmatrix} 8 \\ 8 \\ -12 \end{pmatrix}$

Would you like any more details or further explanation?

Here are five related questions to deepen your understanding:

1. How do you compute the dot product of two vectors?
2. What is the difference between vector addition and scalar multiplication?
3. What is a zero vector, and how does it affect vector operations?
4. How do vectors relate to linear transformations in matrices?
5. How does scalar multiplication affect the direction and magnitude of a vector?

Tip: When performing vector operations, always handle scalar multiplication before adding the vectors.