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

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

Sagot :

GinnyAnsweridn
To solve this problem, let's calculate the magnitudes of each vector operation.

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

We need to determine the magnitudes of the following vector operations and arrange them in ascending order:

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

The calculated magnitudes for these operations are:

1. The magnitude of [tex]\( -\frac{1}{2} \mathbf{u} + 5 \mathbf{v} \)[/tex] is approximately [tex]\( 37.23 \)[/tex].
2. The magnitude of [tex]\( \frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w}) \)[/tex] is approximately [tex]\( 3.89 \)[/tex].
3. The magnitude of [tex]\( \frac{5}{2} \mathbf{u} - 3 \mathbf{w} \)[/tex] is approximately [tex]\( 42.04 \)[/tex].
4. The magnitude of [tex]\( \mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w} \)[/tex] is approximately [tex]\( 28.50 \)[/tex].
5. The magnitude of [tex]\( -4 \mathbf{u} + \frac{1}{2} \mathbf{w} \)[/tex] is approximately [tex]\( 38.71 \)[/tex].

Arranging these in ascending order:

1. [tex]\( \frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w}) \)[/tex] \approx [tex]\( 3.89 \)[/tex]
2. [tex]\( \mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w} \)[/tex] \approx [tex]\( 28.50 \)[/tex]
3. [tex]\( -\frac{1}{2} \mathbf{u} + 5 \mathbf{v} \)[/tex] \approx [tex]\( 37.23 \)[/tex]
4. [tex]\( -4 \mathbf{u} + \frac{1}{2} \mathbf{w} \)[/tex] \approx [tex]\( 38.71 \)[/tex]
5. [tex]\( \frac{5}{2} \mathbf{u} - 3 \mathbf{w} \)[/tex] \approx [tex]\( 42.04 \)[/tex]

So, the correct arrangement of the vector operations in ascending order of their magnitudes is:

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