Sure, let's solve this problem step-by-step.
We are given two vectors:
[tex]\[
\vec{u} = (1, -3, 2) \quad \text{and} \quad \vec{v} = (-1, 4, 3)
\][/tex]
And we need to compute:
[tex]\[
\frac{1}{2} \vec{u} - 3 \vec{v}
\][/tex]
Let's break this down into smaller steps:
### Step 1: Compute [tex]\(\frac{1}{2} \vec{u}\)[/tex]
To find [tex]\(\frac{1}{2} \vec{u}\)[/tex], we need to multiply each component of [tex]\(\vec{u}\)[/tex] by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} \times \vec{u} = \left(\frac{1}{2} \times 1, \frac{1}{2} \times -3, \frac{1}{2} \times 2\right)
\][/tex]
Thus:
[tex]\[
\frac{1}{2} \times \vec{u} = \left(0.5, -1.5, 1.0\right)
\][/tex]
### Step 2: Compute [tex]\(3 \vec{v}\)[/tex]
Next, we multiply each component of [tex]\(\vec{v}\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
3 \times \vec{v} = (3 \times -1, 3 \times 4, 3 \times 3)
\][/tex]
Thus:
[tex]\[
3 \times \vec{v} = (-3, 12, 9)
\][/tex]
### Step 3: Compute [tex]\(\frac{1}{2} \vec{u} - 3 \vec{v}\)[/tex]
Now we subtract the vector [tex]\(3 \vec{v}\)[/tex] from [tex]\(\frac{1}{2} \vec{u}\)[/tex]:
[tex]\[
\frac{1}{2} \vec{u} - 3 \vec{v} = (0.5, -1.5, 1.0) - (-3, 12, 9)
\][/tex]
Subtracting two vectors involves subtracting their respective components:
[tex]\[
(0.5 - (-3), -1.5 - 12, 1.0 - 9)
\][/tex]
Simplifying each component:
[tex]\[
(0.5 + 3, -1.5 - 12, 1.0 - 9) = (3.5, -13.5, -8.0)
\][/tex]
### Final Result
So, the final result of the expression [tex]\(\frac{1}{2} \vec{u} - 3 \vec{v}\)[/tex] is:
[tex]\[
\boxed{(3.5, -13.5, -8.0)}
\][/tex]
Additionally, the intermediate results are:
- [tex]\(\frac{1}{2} \vec{u} = (0.5, -1.5, 1.0)\)[/tex]
- [tex]\(3 \vec{v} = (-3, 12, 9)\)[/tex]
