IDNLearn.com makes it easy to find the right answers to your questions. Find the answers you need quickly and accurately with help from our knowledgeable and dedicated community members.
Sagot :
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
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
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.