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The problem asks to calculate the coordinates of the vector \( 3\vec{u} - 5\vec{v} \), given the vectors:

\[
\vec{u} = (-2, 3, 1) \quad \text{and} \quad \vec{v} = (4, 3, -3)
\]

### Step 1: Multiply each vector by its scalar.

#### For \( 3\vec{u} \):
\[
3\vec{u} = 3 \cdot (-2, 3, 1) = (-6, 9, 3)
\]

#### For \( 5\vec{v} \):
\[
5\vec{v} = 5 \cdot (4, 3, -3) = (20, 15, -15)
\]

### Step 2: Subtract the vectors \( 3\vec{u} \) and \( 5\vec{v} \):
\[
3\vec{u} - 5\vec{v} = (-6, 9, 3) - (20, 15, -15)
\]

Subtracting component-wise:

\[
( -6 - 20, 9 - 15, 3 - (-15) ) = (-26, -6, 18)
\]

### Final answer:
The coordinates of \( 3\vec{u} - 5\vec{v} \) are \( (-26, -6, 18) \).

Would you like any further details on this problem?

Here are 5 related questions to expand on this:
1. What are the geometric interpretations of vector addition and subtraction?
2. How do scalar multiplications affect the direction and magnitude of a vector?
3. How can we use this result to find the angle between the vectors \( \vec{u} \) and \( \vec{v} \)?
4. Can you apply the dot product formula to find whether \( \vec{u} \) and \( \vec{v} \) are orthogonal?
5. How does this problem relate to transformations in 3D space?

**Tip:** Always break vector problems into smaller steps, like scaling and adding components, to avoid confusion.

Dados los vectores \( \vec{u} = (-2, 3, 1) \) y \( \vec{v} = (4, 3, -3) \), elegí la única opción que expresa las coordenadas de \( 3\vec{u} - 5\vec{v} \).

Learn how to calculate the coordinates of the vector expression 3u - 5v, where u = (-2, 3, 1) and v = (4, 3, -3). This step-by-step solution explains scalar multiplication and vector subtraction, essential for vector algebra in Grades 10-12.

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