. Let u = (1, −3, 3), w = (2, 1, −2), and 3u − 2v + 4w = (−1, 2, −3). Find
(a) the vector v.
(b) the linear combination: 3u − 5v + 2w

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?

---

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.

Explore how to solve vector equations using vectors u, v, and w. Learn the step-by-step process to find vector v and compute the linear combination 3u - 5v + 2w. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation