To find the value of $a$, let's use the fact that two vectors are perpendicular if their dot product is zero.

We have the two vectors: $\vec{u} = 2i + 5j - ak$ and $\vec{v} = (a - 3)i + 3j - 4k.$

The dot product of $\vec{u}$ and $\vec{v}$ is: $\vec{u} \cdot \vec{v} = (2)(a - 3) + (5)(3) + (-a)(-4).$

Expanding this, we get: $2(a - 3) + 15 + 4a = 0.$

Now, let's simplify this equation step-by-step.

1. Distribute the $2$ in the first term: $2a - 6 + 15 + 4a = 0.$

2. Combine like terms: $6a + 9 = 0.$

3. Isolate $a$: $6a = -9,$ $a = -\frac{3}{2}.$

Thus, the value of $a$ is $-\frac{3}{2}$.

Would you like further details on this solution, or do you have any questions?

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Here are some related questions that build upon this topic:

1. How would we verify if two vectors are parallel instead of perpendicular?
2. What if the vectors were not perpendicular—how would we approach finding the angle between them?
3. What properties of vectors remain unchanged under scalar multiplication?
4. How do we represent a vector's direction using unit vectors?
5. What conditions must hold for three vectors to be coplanar?

Tip: In vector problems, perpendicularity and parallelism often simplify calculations by allowing us to focus on dot and cross products respectively.