Let's solve the problem step by step:
Given that the magnitude of vector [tex]\(-3 w\)[/tex] is [tex]\(\|-3 w\| = 15\)[/tex], we can start by recalling the property of vector magnitudes:
[tex]\[
\|-3 w\| = 3 \|w\|
\][/tex]
We know that:
[tex]\[
3 \|w\| = 15
\][/tex]
Solving for [tex]\(\|w\|\)[/tex] gives:
[tex]\[
\|w\| = \frac{15}{3} = 5
\][/tex]
Thus, we are looking for vectors [tex]\(w\)[/tex] whose magnitudes are 5.
Now, we examine the magnitude of each given vector to see if it is 5.
1. For the vector [tex]\( \langle 1, -9 \rangle \)[/tex]:
[tex]\[
\| \langle 1, -9 \rangle \| = \sqrt{1^2 + (-9)^2} = \sqrt{1 + 81} = \sqrt{82} \approx 9.06 \neq 5
\][/tex]
2. For the vector [tex]\( \langle -3, 4 \rangle \)[/tex]:
[tex]\[
\| \langle -3, 4 \rangle \| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
3. For the vector [tex]\( \langle 4, 5 \rangle \)[/tex]:
[tex]\[
\| \langle 4, 5 \rangle \| = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.40 \neq 5
\][/tex]
4. For the vector [tex]\( \langle -5, -3 \rangle \)[/tex]:
[tex]\[
\| \langle -5, -3 \rangle \| = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 \neq 5
\][/tex]
5. For the vector [tex]\( \langle 0, -5 \rangle \)[/tex]:
[tex]\[
\| \langle 0, -5 \rangle \| = \sqrt{0^2 + (-5)^2} = \sqrt{0 + 25} = \sqrt{25} = 5
\][/tex]
Given this analysis, the vectors that have a magnitude of 5 and are possible representations of vector [tex]\(w\)[/tex] are:
[tex]\[
\langle -3, 4 \rangle \text{ and } \langle 0, -5 \rangle
\][/tex]
Therefore, the correct answers are:
- [tex]\(\langle -3, 4 \rangle\)[/tex]
- [tex]\(\langle 0, -5 \rangle\)[/tex]
