To determine which components are a possible representation of vector [tex]\( u \)[/tex] given that [tex]\( \|-4u\| \approx 14.42 \)[/tex], let's work through the calculations step-by-step.
First, we need to understand the relationship given in the question:
[tex]\[
\|-4u\| \approx 14.42
\][/tex]
This means that the magnitude (or norm) of [tex]\(-4\)[/tex] times the vector [tex]\( u \)[/tex] is approximately 14.42.
The magnitude of a vector [tex]\( v = \)[/tex] is calculated using the formula:
[tex]\[
\|v\| = \sqrt{v_1^2 + v_2^2}
\][/tex]
Given these choices, let's evaluate the magnitude for each vector option [tex]\( u \)[/tex]. Notice that scaling a vector by [tex]\(-4\)[/tex] affects its magnitude linearly:
[tex]\[
|-4| \cdot \|u\| = 4 \cdot \|u\|
\][/tex]
Let's set this equal to the given approximation:
[tex]\[
4 \cdot \|u\| = 14.42
\][/tex]
Solving for [tex]\(\|u\|\)[/tex]:
[tex]\[
\|u\| = \frac{14.42}{4} = 3.605
\][/tex]
Next, we need to check which of the provided vectors [tex]\( u \)[/tex] has a magnitude approximately equal to 3.605.
Option A: [tex]\( <3, -2> \)[/tex]
[tex]\[
\|<3, -2>\| = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.605
\][/tex]
This is a potential candidate as the magnitude is approximately 3.605.
Option B: [tex]\( <-1, 4> \)[/tex]
[tex]\[
\|<-1, 4>\| = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.123
\][/tex]
This magnitude is not approximately equal to 3.605.
Option C: [tex]\( <2, 2> \)[/tex]
[tex]\[
\|<2, 2>\| = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.828
\][/tex]
This magnitude is not approximately equal to 3.605.
Option D: [tex]\( <-2, -4> \)[/tex]
[tex]\[
\|<-2, -4>\| = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.472
\][/tex]
This magnitude is not approximately equal to 3.605.
After evaluating each option, the vector [tex]\( <3, -2> \)[/tex] is the correct choice because its magnitude matches our requirement:
[tex]\[
\|4u\| \approx 14.42 \implies u = <3, -2>
\][/tex]
Thus, the correct answer is:
A. [tex]\( <3, -2> \)[/tex]
