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Let's break down the problem. You need to calculate the cross product \(( -2\mathbf{u} - 3\mathbf{w} ) \times \mathbf{w}\), given that \(\mathbf{u} \times \mathbf{w} = (-5, -1, 1)\).

### Step 1: Express \(\mathbf{v} = -2\mathbf{u} - 3\mathbf{w}\)
We don't have explicit vectors \(\mathbf{u}\) and \(\mathbf{w}\), but we can use the cross product property:
\[
(\mathbf{v} \times \mathbf{w}) = (-2\mathbf{u} - 3\mathbf{w}) \times \mathbf{w}
\]

### Step 2: Distribute the cross product using linearity
Using the distributive property:
\[
(\mathbf{v} \times \mathbf{w}) = (-2\mathbf{u} \times \mathbf{w}) - (3\mathbf{w} \times \mathbf{w})
\]
Where:
- \(\mathbf{u} \times \mathbf{w}\) is given as \((-5, -1, 1)\),
- \(\mathbf{w} \times \mathbf{w} = 0\) (since the cross product of any vector with itself is zero).

### Step 3: Compute the cross product
\[
(\mathbf{v} \times \mathbf{w}) = -2(\mathbf{u} \times \mathbf{w}) - 3 \times 0
\]
\[
(\mathbf{v} \times \mathbf{w}) = -2(-5, -1, 1)
\]
\[
(\mathbf{v} \times \mathbf{w}) = (10, 2, -2)
\]

So, the result of the cross product \(( -2\mathbf{u} - 3\mathbf{w} ) \times \mathbf{w}\) is \(\boxed{(10, 2, -2)}\).

Would you like any more details, or do you have any questions about the process?

### Follow-Up Questions:
1. How do you calculate the cross product between two vectors in 3D?
2. Why does the cross product of a vector with itself result in the zero vector?
3. What are some properties of the cross product that can simplify calculations?
4. How does the direction of the cross product relate to the vectors involved?
5. How would the result change if \(\mathbf{u} \times \mathbf{w}\) were different?

### Tip:
Always remember that the cross product is anti-commutative, meaning \(\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})\). This can be useful for simplifying expressions.

Calculate the cross product assuming that u × w = (-5, -1, 1). Find (-2u - 3w) × w.

Learn how to calculate the cross product for the vector expression (-2u - 3w) × w, given that u × w = (-5, -1, 1). This detailed solution covers vector algebra concepts, cross product formulas, and key properties of cross products.

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