Sure! Let's go through the steps necessary to arrange the given vector operations in ascending order of the magnitudes of their resultant vectors.
Given vectors:
- \( u = \langle 9, -2 \rangle \)
- \( v = \langle -1, 7 \rangle \)
- \( w = \langle -5, -8 \rangle \)
We are given five vector operations:
1. \( -\frac{1}{2} u + 5 v \)
2. \( \frac{1}{6} (u + 2 v - w) \)
3. \( \frac{5}{2} u - 3 w \)
4. \( -4 v + \frac{1}{2} w \)
5. \( 3 u - v - \frac{5}{2} w \)
To compare these, let's find their magnitudes and determine their order.
### 1. Magnitude of \( -\frac{1}{2} u + 5 v \)
Resultant vector: \(\langle -51/2, 177/2 \rangle \)
Magnitude: \( 37.2324 \)
### 2. Magnitude of \( \frac{1}{6} (u + 2 v - w) \)
Resultant vector: \(\langle 1, 8/3 \rangle \)
Magnitude: \( 3.8873 \)
### 3. Magnitude of \( \frac{5}{2} u - 3 w \)
Resultant vector: \(\langle 67, -19 \rangle \)
Magnitude: \( 42.0387 \)
### 4. Magnitude of \( -4 v + 0.5 w \)
Resultant vector: \(\langle -3/2, -38 \rangle \)
Magnitude: \( 32.0351 \)
### 5. Magnitude of \( 3 u - v - \frac{5}{2} w \)
Resultant vector: \(\langle 89.5, -17.5 \rangle \)
Magnitude: \( 41.1005 \)
### Ordering by Magnitude
We place the vector operations in ascending order of their magnitudes:
1. \( \frac{1}{6} (u + 2 v - w) \) with magnitude \( 3.8873 \)
2. \( -4 v + \frac{1}{2} w \) with magnitude \( 32.0351 \)
3. \( -\frac{1}{2} u + 5 v \) with magnitude \( 37.2324 \)
4. \( 3 u - v - \frac{5}{2} w \) with magnitude \( 41.1005 \)
5. \( \frac{5}{2} u - 3 w \) with magnitude \( 42.0387 \)
Thus, the vector operations in ascending order of the magnitudes of their resultant vectors are:
\[
\begin{array}{c}
\\
\frac{1}{6}(u+2 v-w) \\
\\
\hline
\\
-4 v+\frac{1}{2} w \\
\\
\hline
\\
-\frac{1}{2} u+5 v \\
\\
\hline
\\
3 u-v-\frac{5}{2} w \\
\\
\hline
\\
\frac{5}{2} u-3 w \\
\\
\end{array}
