The problem involves performing some mathematical operations with vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$, as well as finding $\alpha$. Let's analyze and solve step by step.

Extracted information from the image:

1. Vectors are given as: $\vec{a} = (2, -1, 0), \quad \vec{b} = (7i, 0, i - 3), \quad \vec{c} = (0, 2, 3)$

2. An operation is mentioned involving $\alpha (2c - 6)$.

Step 1: Simplify $2\vec{c} - 6$:

$2\vec{c} = 2 \cdot (0, 2, 3) = (0, 4, 6)$ $2\vec{c} - 6 = (0, 4, 6) - (6, 6, 6) = (-6, -2, 0)$

Step 2: Multiply by $\alpha$:

$\alpha (2\vec{c} - 6) = \alpha \cdot (-6, -2, 0) = (-6\alpha, -2\alpha, 0)$

If you'd like, I can expand further on vector operations, solve for $\alpha$ if needed, or clarify additional steps. Let me know!

Related Questions:

1. How do we compute scalar multiplication of vectors step by step?
2. What happens if $\alpha$ is given a specific value?
3. Can we determine the magnitude of these vectors?
4. How do you handle vectors involving imaginary components like $\vec{b}$?
5. How does subtraction of vectors operate geometrically?

Tip: Always verify the components of your vectors before performing operations to avoid calculation errors.