To determine the reflection of the vector [tex]\(\langle -1, 5 \rangle\)[/tex] across the line [tex]\(x = y\)[/tex], we follow a straightforward transformation. When a vector is reflected across the line [tex]\(x = y\)[/tex], the [tex]\(x\)[/tex]-coordinate and the [tex]\(y\)[/tex]-coordinate of the vector are swapped.
We start with the vector:
[tex]\[
\langle -1, 5 \rangle
\][/tex]
To reflect this vector across the line [tex]\(x = y\)[/tex], we swap the components:
[tex]\[
\langle -1, 5 \rangle \quad \rightarrow \quad \langle 5, -1 \rangle
\][/tex]
Thus, the vector [tex]\(\langle -1, 5 \rangle\)[/tex] reflected across the line [tex]\(x = y\)[/tex] becomes [tex]\(\langle 5, -1 \rangle\)[/tex].
Now, we need to select the correct answer from the given options. The correct reflected vector is:
[tex]\[
\left[\begin{array}{c}5 \\ -1\end{array}\right]
\][/tex]
So, the correct answer is:
C. [tex]\(\left[\begin{array}{c}5 \\ -1\end{array}\right]\)[/tex]
