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Drag each tile to the correct box.

Vector [tex]u = \langle 9, -2 \rangle[/tex], [tex]v = \langle -1, 7 \rangle[/tex], and [tex]w = \langle -5, -8 \rangle[/tex].

Arrange the vector operations in ascending order of the magnitudes of their resultant vectors.

[tex]\[
\begin{array}{c}
-\frac{1}{2} u + 5 v \\
\frac{1}{6}(u + 2 v - w) \\
\hline \frac{5}{2} u - 3 w \\
u - \frac{3}{2} v + 2 w \\
-4 v + \frac{1}{2} w \\
3 u - v - \frac{5}{2} w \\
\hline
\end{array}
\][/tex]

[tex]\[
\begin{array}{c}
\square \\
\square \\
\square \\
\square \\
\square \\
\square \\
\end{array}
\][/tex]



Answer :

GinnyAnswer
Let's arrange the given vector operations in ascending order based on the magnitudes of their resultant vectors.

Given vectors:
- [tex]\( u = \langle 9, -2 \rangle \)[/tex]
- [tex]\( v = \langle -1, 7 \rangle \)[/tex]
- [tex]\( w = \langle -5, -8 \rangle \)[/tex]

Vector operations:
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]\( u - \frac{3}{2} v + 2 w \)[/tex]
5. [tex]\( -4 v + \frac{1}{2} w \)[/tex]
6. [tex]\( 3 u - v - \frac{5}{2} w \)[/tex]

Corresponding magnitudes of resultant vectors:
1. [tex]\( -\frac{1}{2} u + 5 v \approx 37.23 \)[/tex]
2. [tex]\( \frac{1}{6} (u + 2 v - w) \approx 3.89 \)[/tex]
3. [tex]\( \frac{5}{2} u - 3 w \approx 42.04 \)[/tex]
4. [tex]\( u - \frac{3}{2} v + 2 w \approx 28.50 \)[/tex]
5. [tex]\( -4 v + \frac{1}{2} w \approx 32.04 \)[/tex]
6. [tex]\( 3 u - v - \frac{5}{2} w \approx 41.10 \)[/tex]

Arranging these magnitudes in ascending order:

1. [tex]\( \frac{1}{6} (u + 2 v - w) \approx 3.89 \)[/tex]
2. [tex]\( u - \frac{3}{2} v + 2 w \approx 28.50 \)[/tex]
3. [tex]\( -4 v + \frac{1}{2} w \approx 32.04 \)[/tex]
4. [tex]\( -\frac{1}{2} u + 5 v \approx 37.23 \)[/tex]
5. [tex]\( 3 u - v - \frac{5}{2} w \approx 41.10 \)[/tex]
6. [tex]\( \frac{5}{2} u - 3 w \approx 42.04 \)[/tex]

Therefore, the correct order from smallest to largest magnitude is:

1. [tex]\( \frac{1}{6} (u + 2 v - w) \)[/tex] (Smallest)
2. [tex]\( u - \frac{3}{2} v + 2 w \)[/tex]
3. [tex]\( -4 v + \frac{1}{2} w \)[/tex]
4. [tex]\( -\frac{1}{2} u + 5 v \)[/tex]
5. [tex]\( 3 u - v - \frac{5}{2} w \)[/tex]
6. [tex]\( \frac{5}{2} u - 3 w \)[/tex] (Largest)

The arrangement of the tiles should be:

[tex]\[ \begin{array}{c} \frac{1}{6}(u+2 v-w) \\ u-\frac{3}{2} v+2 w \\ -4 v+\frac{1}{2} w \\ -\frac{1}{2} u+5 v \\ 3 u-v-\frac{5}{2} w \\ \frac{5}{2} u-3 w \\ \end{array} \][/tex]

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