Let's break down the problem step-by-step.
You are given three vectors:
[tex]\[ u = \langle 9, -2 \rangle \][/tex]
[tex]\[ v = \langle -1, 7 \rangle \][/tex]
[tex]\[ w = \langle -5, -8 \rangle \][/tex]
We need to calculate the resultant vectors for each of the given operations, find their magnitudes, and then sort them in ascending order of their magnitudes.
The given vector operations are:
1. [tex]\(-\frac{1}{2} u + 5 v\)[/tex]
2. [tex]\(\frac{1}{6}(u + 2 v - w)\)[/tex]
3. [tex]\(\frac{5}{2} u - 3 w\)[/tex]
4. [tex]\(-4 v + \frac{1}{2} v + 2 w\)[/tex]
5. [tex]\(3 u - v - \frac{5}{2} w\)[/tex]
Based on the provided answer:
The magnitudes of the resultant vectors are:
1. [tex]\[37.23237838226293\][/tex]
2. [tex]\[3.8873012632302\][/tex]
3. [tex]\[42.03867267171979\][/tex]
4. [tex]\[41.018288603987365\][/tex]
5. [tex]\[41.100486615124154\][/tex]
Arranging these magnitudes in ascending order, we get:
2. [tex]\(\frac{1}{6}(u + 2 v - w)\)[/tex]
1. [tex]\(-\frac{1}{2} u + 5 v\)[/tex]
4. [tex]\(-4 v + \frac{1}{2} v + 2 w\)[/tex]
5. [tex]\(3 u - v - \frac{5}{2} w\)[/tex]
3. [tex]\(\frac{5}{2} u - 3 w\)[/tex]
So, the correct order of the vector operations in ascending order of their magnitudes is:
[tex]\[ \frac{1}{6}(u + 2 v - w) \][/tex]
[tex]\[ -\frac{1}{2} u + 5 v \][/tex]
[tex]\[ -4 v + \frac{1}{2} v + 2 w \][/tex]
[tex]\[ 3 u - v - \frac{5}{2} w \][/tex]
[tex]\[ \frac{5}{2} u - 3 w \][/tex]
Thus, the arrangement in the boxes should be:
[tex]$
\begin{array}{c}
\frac{1}{6}(u + 2 v - w) \\
-\frac{1}{2} u + 5 v \\
-4 v + \frac{1}{2} v + 2 w \\
3 u - v - \frac{5}{2} w \\
\frac{5}{2} u - 3 w \\
\end{array}
$[/tex]
