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Problem 1:

Given the vectors $a = (1, 2, 3)$, $b = (2, -3, 1)$, and $c = (3, 2, -1)$, we need to find:

a) $a - c$

To compute $a - c$, subtract each corresponding component of vector $c$ from vector $a$: $a - c = (1 - 3, 2 - 2, 3 - (-1)) = (-2, 0, 4)$

b) $3b - (a + 2c)$

First, we need to calculate $a + 2c$: $2c = 2 \times (3, 2, -1) = (6, 4, -2)$ $a + 2c = (1, 2, 3) + (6, 4, -2) = (7, 6, 1)$

Now compute $3b - (a + 2c)$: $3b = 3 \times (2, -3, 1) = (6, -9, 3)$ $3b - (a + 2c) = (6, -9, 3) - (7, 6, 1) = (-1, -15, 2)$

Problem 2:

Given vectors $a = (3, -2, 5)$, $b = (1, -3, -1)$, and $c = (1, -1, 3)$, we are asked to find the vector $u$ that satisfies the equation: $2a + u + c = 3u + b$

First, rearrange the equation: $2a + c - b = 2u$

Now, substitute the values of the vectors: $2a = 2 \times (3, -2, 5) = (6, -4, 10)$ $2a + c = (6, -4, 10) + (1, -1, 3) = (7, -5, 13)$ $2a + c - b = (7, -5, 13) - (1, -3, -1) = (6, -2, 14)$

Thus, the equation becomes: $2u = (6, -2, 14)$ Finally, solve for $u$: $u = \frac{1}{2} \times (6, -2, 14) = (3, -1, 7)$

Summary of Answers:

1. a) $a - c = (-2, 0, 4)$ b) $3b - (a + 2c) = (-1, -15, 2)$
2. $u = (3, -1, 7)$

Do you need further details on any of the steps? Let me know!

Here are some additional questions to further explore these concepts:

1. How do vector addition and subtraction work in higher dimensions?
2. What is the significance of scaling a vector, such as in the operation $3b$?
3. How would you solve vector equations with more than three vectors?
4. What other vector operations are commonly used, such as the dot product or cross product?
5. How can you use the concept of vector spaces in solving systems of linear equations?

Tip: When working with vector operations, always keep track of the components, and ensure that you follow the correct order of operations to avoid mistakes!