To solve this problem, we need to match each vector subtraction with the corresponding magnitude. Here are the steps involved:
1. Vector Subtraction: First, we subtract the given vectors to find the resulting vectors.
2. Magnitude Calculation: Compute the magnitudes of these resulting vectors using the Euclidean distance formula [tex]\( \| \mathbf{a} - \mathbf{b} \| = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2} \)[/tex].
Given vectors:
- [tex]\( \mathbf{u} = \langle -1, -3 \rangle \)[/tex]
- [tex]\( \mathbf{v} = \langle 5, -8 \rangle \)[/tex]
- [tex]\( \mathbf{w} = \langle 5, -2 \rangle \)[/tex]
- [tex]\( \mathbf{z} = \langle 3, 1 \rangle \)[/tex]
Subtractions and Magnitudes:
1. Subtraction [tex]\( \mathbf{u} - \mathbf{v} \)[/tex]:
- Resulting vector: [tex]\( \langle -1 - 5, -3 - (-8) \rangle = \langle -6, 5 \rangle \)[/tex]
- Magnitude: [tex]\( \|\mathbf{u} - \mathbf{v}\| = 7.81 \)[/tex]
2. Subtraction [tex]\( \mathbf{u} - \mathbf{w} \)[/tex]:
- Resulting vector: [tex]\( \langle -1 - 5, -3 - (-2) \rangle = \langle -6, -1 \rangle \)[/tex]
- Magnitude: [tex]\( \|\mathbf{u} - \mathbf{w}\| = 6.08 \)[/tex]
3. Subtraction [tex]\( \mathbf{u} - \mathbf{z} \)[/tex]:
- Resulting vector: [tex]\( \langle -1 - 3, -3 - 1 \rangle = \langle -4, -4 \rangle \)[/tex]
- Magnitude: [tex]\( \|\mathbf{u} - \mathbf{z}\| = 5.66 \)[/tex]
4. Subtraction [tex]\( \mathbf{v} - \mathbf{w} \)[/tex]:
- Resulting vector: [tex]\( \langle 5 - 5, -8 - (-2) \rangle = \langle 0, -6 \rangle \)[/tex]
- Magnitude: [tex]\( \|\mathbf{v} - \mathbf{w}\| = 6 \)[/tex]
5. Subtraction [tex]\( \mathbf{v} - \mathbf{z} \)[/tex]:
- Resulting vector: [tex]\( \langle 5 - 3, -8 - 1 \rangle = \langle 2, -9 \rangle \)[/tex]
- Magnitude: [tex]\( \|\mathbf{v} - \mathbf{z}\| = 9.22 \)[/tex]
6. Subtraction [tex]\( \mathbf{w} - \mathbf{z} \)[/tex]:
- Resulting vector: [tex]\( \langle 5 - 3, -2 - 1 \rangle = \langle 2, -3 \rangle \)[/tex]
- Magnitude: [tex]\( \|\mathbf{w} - \mathbf{z}\| = 3.61 \)[/tex]
Now, let's match the vector subtractions with their magnitudes:
- [tex]\( \|\mathbf{u} - \mathbf{v}\| = 7.81 \)[/tex]
- [tex]\( \|\mathbf{u} - \mathbf{w}\| = 6.08 \)[/tex]
- [tex]\( \|\mathbf{u} - \mathbf{z}\| = 5.66 \)[/tex]
- [tex]\( \|\mathbf{v} - \mathbf{w}\| = 6 \)[/tex]
- [tex]\( \|\mathbf{v} - \mathbf{z}\| = 9.22 \)[/tex]
- [tex]\( \|\mathbf{w} - \mathbf{z}\| = 3.61 \)[/tex]
Thus, pairs are:
- [tex]\( \|\mathbf{u} - \mathbf{v}\| \)[/tex] matches with 7.81
- [tex]\( \|\mathbf{u} - \mathbf{w}\| \)[/tex] matches with 6.08
- [tex]\( \|\mathbf{u} - \mathbf{z}\| \)[/tex] matches with 5.66
- [tex]\( \|\mathbf{v} - \mathbf{w}\| \)[/tex] matches with 6
