Let's solve the problem step-by-step:
First, let's identify the given vectors:
- [tex]\( u = \langle -1, -3 \rangle \)[/tex]
- [tex]\( v = \langle 5, -8 \rangle \)[/tex]
- [tex]\( w = \langle 5, -2 \rangle \)[/tex]
- [tex]\( z = \langle \langle, 1 \rangle \)[/tex]
Next, we need to compute the results of the following vector subtractions and then determine which magnitude corresponds to each subtraction:
1. [tex]\( \|u - z\| \)[/tex]
2. [tex]\( \|v - w\| \)[/tex]
3. [tex]\( \|w - u\| \)[/tex]
4. [tex]\( \|v - u\| \)[/tex]
We have the magnitudes specified:
- 4.12
- 6.0
- 6.08
- 7.81
Now let's match each magnitude with its corresponding vector subtraction result:
1. [tex]\( \|u - z\| = 4.12 \)[/tex]
2. [tex]\( \|v - w\| = 6.0 \)[/tex]
3. [tex]\( \|w - u\| = 6.08 \)[/tex]
4. [tex]\( \|v - u\| = 7.81 \)[/tex]
Based on this information, we can now fill in the boxes correctly:
[tex]\[
\|u-z\| \longrightarrow 4.12
\][/tex]
[tex]\(\|v-w\| \longrightarrow 6.0\)[/tex]
[tex]\(\|w-u\| \longrightarrow 6.08\)[/tex]
[tex]\(\|v-u\| \longrightarrow 7.81\)[/tex]
Thus, the completed array of matches is:
```
Vector Subtractions
\|u-z\| --------------- 4.12
\|v-w\| --------------- 6.0
\|w-u\| --------------- 6.08
\|v-u\| --------------- 7.81
```
