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To solve the problem of determining the values of \(a\) and \(b\) for which the two vectors \(\vec{u} = (2a - b, -3, a - 15)\) and \(\vec{v} = (5, 3a, -1)\) are perpendicular, we use the fact that two vectors are perpendicular if and only if their dot product is zero.

The dot product of \(\vec{u}\) and \(\vec{v}\) is given by:

\[
\vec{u} \cdot \vec{v} = (2a - b) \cdot 5 + (-3) \cdot (3a) + (a - 15) \cdot (-1)
\]

Expanding this expression:

\[
\vec{u} \cdot \vec{v} = 5(2a - b) + (-3)(3a) + (-1)(a - 15)
\]

Simplifying each term:

\[
\vec{u} \cdot \vec{v} = 10a - 5b - 9a - a + 15
\]

Combining like terms:

\[
\vec{u} \cdot \vec{v} = (10a - 9a - a) - 5b + 15 = 0a - 5b + 15
\]

This simplifies to:

\[
-5b + 15 = 0
\]

Solving for \(b\):

\[
5b = 15
\]
\[
b = 3
\]

Thus, the value of \(b\) is 3. The parameter \(a\) does not appear in the final equation, meaning that \(a\) can be any value.

Therefore, the correct option is:

\[
\boxed{e: a \text{ can be any value}, b = 3}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you calculate the dot product of two vectors in three dimensions?
2. What is the geometric interpretation of two vectors being perpendicular?
3. How would the solution change if the vectors were parallel instead of perpendicular?
4. Can you derive a similar condition for vectors in higher dimensions?
5. How do different values of \(a\) affect the direction of vector \(\vec{u}\)?

**Tip:** When working with vectors, always check if the conditions of the problem relate to the dot product, especially for perpendicularity (orthogonality).

If the two vectors u = (2a - b, -3, a - 15) and v = (5, 3a, -1) are perpendicular, determine the values of a and b.

This problem involves finding the values of a and b such that two vectors are perpendicular. The solution uses the dot product formula and simplifies the expressions to determine the parameters a and b. Suitable for high school students studying vector algebra.

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