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Sagot :
To arrange the given vector operations in ascending order of the magnitudes of their resultant vectors, we'll determine the magnitudes of each vector operation and then sort them accordingly.
The given vectors:
[tex]\[ \mathbf{u} = \langle 9, -2 \rangle, \mathbf{v} = \langle -1, 7 \rangle, \mathbf{w} = \langle -5, -8 \rangle \][/tex]
The operations and their corresponding magnitudes:
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]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex]
6. [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex]
Here are the magnitudes in ascending order:
1. Magnitude of [tex]\(\frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w})\)[/tex] is approximately [tex]\(3.8873\)[/tex]
2. Magnitude of [tex]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex] is approximately [tex]\(28.5044\)[/tex]
3. Magnitude of [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex] is approximately [tex]\(32.0351\)[/tex]
4. Magnitude of [tex]\(\mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w}\)[/tex] is approximately [tex]\(37.2324\)[/tex]
5. Magnitude of [tex]\(\frac{5}{2} \mathbf{u} - 3 \mathbf{w}\)[/tex] is approximately [tex]\(41.1005\)[/tex]
6. Magnitude of [tex]\(-\frac{1}{2} \mathbf{u} + 5 \mathbf{v}\)[/tex] is approximately [tex]\(42.0387\)[/tex]
Therefore, the correct order of operations in ascending order of their magnitudes is:
1. [tex]\(\frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w})\)[/tex]
2. [tex]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex]
3. [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex]
4. [tex]\(\mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w}\)[/tex]
5. [tex]\(\frac{5}{2} \mathbf{u} - 3 \mathbf{w}\)[/tex]
6. [tex]\(-\frac{1}{2} \mathbf{u} + 5 \mathbf{v}\)[/tex]
So, the tiles should be arranged in the following order from left to right:
[tex]\(\frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w})\)[/tex], [tex]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex], [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex], [tex]\(\mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w}\)[/tex], [tex]\(\frac{5}{2} \mathbf{u} - 3 \mathbf{w}\)[/tex], [tex]\(-\frac{1}{2} \mathbf{u} + 5 \mathbf{v}\)[/tex].
The given vectors:
[tex]\[ \mathbf{u} = \langle 9, -2 \rangle, \mathbf{v} = \langle -1, 7 \rangle, \mathbf{w} = \langle -5, -8 \rangle \][/tex]
The operations and their corresponding magnitudes:
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]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex]
6. [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex]
Here are the magnitudes in ascending order:
1. Magnitude of [tex]\(\frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w})\)[/tex] is approximately [tex]\(3.8873\)[/tex]
2. Magnitude of [tex]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex] is approximately [tex]\(28.5044\)[/tex]
3. Magnitude of [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex] is approximately [tex]\(32.0351\)[/tex]
4. Magnitude of [tex]\(\mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w}\)[/tex] is approximately [tex]\(37.2324\)[/tex]
5. Magnitude of [tex]\(\frac{5}{2} \mathbf{u} - 3 \mathbf{w}\)[/tex] is approximately [tex]\(41.1005\)[/tex]
6. Magnitude of [tex]\(-\frac{1}{2} \mathbf{u} + 5 \mathbf{v}\)[/tex] is approximately [tex]\(42.0387\)[/tex]
Therefore, the correct order of operations in ascending order of their magnitudes is:
1. [tex]\(\frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w})\)[/tex]
2. [tex]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex]
3. [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex]
4. [tex]\(\mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w}\)[/tex]
5. [tex]\(\frac{5}{2} \mathbf{u} - 3 \mathbf{w}\)[/tex]
6. [tex]\(-\frac{1}{2} \mathbf{u} + 5 \mathbf{v}\)[/tex]
So, the tiles should be arranged in the following order from left to right:
[tex]\(\frac{1}{6} (\mathbf{u} + 2 \mathbf{v} - \mathbf{w})\)[/tex], [tex]\(-4 \mathbf{v} + \frac{1}{2} \mathbf{w}\)[/tex], [tex]\(3 \mathbf{u} - \mathbf{v} - \frac{5}{2} \mathbf{w}\)[/tex], [tex]\(\mathbf{u} - \frac{3}{2} \mathbf{v} + 2 \mathbf{w}\)[/tex], [tex]\(\frac{5}{2} \mathbf{u} - 3 \mathbf{w}\)[/tex], [tex]\(-\frac{1}{2} \mathbf{u} + 5 \mathbf{v}\)[/tex].
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