Given the vectors $u = (1, -3, 3)$, $w = (2, 1, -2)$, and the equation $3u - 2v + 4w = (-1, 2, -3)$, we need to solve the following:

Part (a) - Finding the vector $v$

We start with the equation: $3u - 2v + 4w = (-1, 2, -3)$

Substitute $u = (1, -3, 3)$ and $w = (2, 1, -2)$:

$3(1, -3, 3) - 2v + 4(2, 1, -2) = (-1, 2, -3)$

First, calculate $3u$ and $4w$:

$3u = 3(1, -3, 3) = (3, -9, 9)$

$4w = 4(2, 1, -2) = (8, 4, -8)$

Now substitute these into the equation:

$(3, -9, 9) - 2v + (8, 4, -8) = (-1, 2, -3)$

Add the vectors $3u$ and $4w$:

$(3 + 8, -9 + 4, 9 - 8) = (11, -5, 1)$

So the equation becomes:

$11 - 2v = (-1, 2, -3)$

Now, isolate $2v$ by subtracting $(11, -5, 1)$ from both sides:

$-2v = (-1, 2, -3) - (11, -5, 1)$

$-2v = (-1 - 11, 2 + 5, -3 - 1) = (-12, 7, -4)$

Finally, solve for $v$ by dividing each component by -2:

$v = \left(\frac{-12}{-2}, \frac{7}{-2}, \frac{-4}{-2}\right) = (6, -\frac{7}{2}, 2)$

So, the vector $v$ is:

$\boxed{v = \left(6, -\frac{7}{2}, 2\right)}$

Part (b) - Finding the linear combination $3u - 5v + 2w$

Now we calculate the linear combination $3u - 5v + 2w$.

Substitute the values of $u$, $v$, and $w$:

$3u = (3, -9, 9)$

$5v = 5\left(6, -\frac{7}{2}, 2\right) = \left(30, -\frac{35}{2}, 10\right)$

$2w = 2(2, 1, -2) = (4, 2, -4)$

Now, compute the linear combination:

$3u - 5v + 2w = (3, -9, 9) - \left(30, -\frac{35}{2}, 10\right) + (4, 2, -4)$

Subtract $5v$ from $3u$:

$(3 - 30, -9 + \frac{35}{2}, 9 - 10) = \left(-27, -\frac{18}{2} + \frac{35}{2}, -1\right) = \left(-27, \frac{17}{2}, -1\right)$

Finally, add $2w$:

$(-27 + 4, \frac{17}{2} + 2, -1 - 4) = \left(-23, \frac{17}{2} + \frac{4}{2}, -5\right) = \left(-23, \frac{21}{2}, -5\right)$

So the linear combination $3u - 5v + 2w$ is:

$\boxed{\left(-23, \frac{21}{2}, -5\right)}$

Would you like more details or have any questions?

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Here are five related questions to explore:

1. How would you find $2u + v - w$ given the vectors $u$, $v$, and $w$?
2. What is the dot product of vectors $u$ and $v$?
3. How do you determine if vectors $u$ and $w$ are orthogonal?
4. Can you find the magnitude of the vector $v$ found in part (a)?
5. How would you calculate the cross product of vectors $u$ and $w$?

Tip: When solving vector equations, it's helpful to work systematically through each operation to avoid errors and ensure clarity.