To find the correct vectors [tex]\( \mathbf{w} \)[/tex] such that the magnitude of the vector [tex]\( -3\mathbf{w} \)[/tex] is 15, we can follow these steps:
1. Determine the magnitude condition for [tex]\( \mathbf{w} \)[/tex]:
We know that the magnitude of [tex]\( -3\mathbf{w} \)[/tex] is given by [tex]\(\|-3\mathbf{w}\| = 15\)[/tex].
2. Using the properties of magnitudes:
Recall that the magnitude of [tex]\( -3\mathbf{w} \)[/tex] can be written as:
[tex]\[
\|-3\mathbf{w}\| = 3 \|\mathbf{w}\|
\][/tex]
Thus:
[tex]\[
3 \|\mathbf{w}\| = 15
\][/tex]
Which simplifies to:
[tex]\[
\|\mathbf{w}\| = 5
\][/tex]
3. Calculate the magnitudes of each given vector:
We need to find which of the given vectors have a magnitude of 5. Let's calculate the magnitudes:
- For [tex]\( \mathbf{w} = \langle 1, -9 \rangle \)[/tex]:
[tex]\[
\|\mathbf{w}\| = \sqrt{1^2 + (-9)^2} = \sqrt{1 + 81} = \sqrt{82}
\][/tex]
- For [tex]\( \mathbf{w} = \langle -3, 4 \rangle \)[/tex]:
[tex]\[
\|\mathbf{w}\| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
- For [tex]\( \mathbf{w} = \langle 4, 5 \rangle \)[/tex]:
[tex]\[
\|\mathbf{w}\| = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}
\][/tex]
- For [tex]\( \mathbf{w} = \langle -5, -3 \rangle \)[/tex]:
[tex]\[
\|\mathbf{w}\| = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}
\][/tex]
- For [tex]\( \mathbf{w} = \langle 0, -5 \rangle \)[/tex]:
[tex]\[
\|\mathbf{w}\| = \sqrt{0^2 + (-5)^2} = \sqrt{0 + 25} = \sqrt{25} = 5
\][/tex]
4. Identify the vectors that match the magnitude condition [tex]\( \|\mathbf{w}\| = 5 \)[/tex]:
From the calculations above, the vectors that have a magnitude of 5 are:
- [tex]\(\langle -3, 4 \rangle \)[/tex]
- [tex]\(\langle 0, -5 \rangle \)[/tex]
Thus, the correct answers are:
[tex]\[ \langle -3, 4 \rangle \][/tex]
[tex]\[ \langle 0, -5 \rangle \][/tex]
